Hippocrates

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hippocrates

 

Hippocrates of Kos
A conventionalized image in a Roman "portrait" bust (19th-century engraving)
Born	c. 460 BC Kos, Ancient Greece
Died	c. 370 BC (aged approximately 90) Larissa, Ancient Greece
Occupation	Physician
Era	Classical Greece

Hippocrates of Kos (/hɪˈpɒkrətiːz/; Greek: Ἱπποκράτης ὁ Κῷος, translit. Hippokrátēs ho Kôios; c. 460 – c. 370 BC), also known as Hippocrates II, was a Greek physician of the classical period who is considered one of the most outstanding figures in the history of medicine. He is traditionally referred to as the "Father of Medicine" in recognition of his lasting contributions to the field, such as the use of prognosis and clinical observation, the systematic categorization of diseases, or the formulation of humoural theory. The Hippocratic school of medicine revolutionized ancient Greek medicine, establishing it as a discipline distinct from other fields with which it had traditionally been associated (theurgy and philosophy), thus establishing medicine as a profession.

However, the achievements of the writers of the Hippocratic Corpus, the practitioners of Hippocratic medicine, and the actions of Hippocrates himself were often conflated; thus very little is known about what Hippocrates actually thought, wrote, and did. Hippocrates is commonly portrayed as the paragon of the ancient physician and credited with coining the Hippocratic Oath, which is still relevant and in use today. He is also credited with greatly advancing the systematic study of clinical medicine, summing up the medical knowledge of previous schools, and prescribing practices for physicians through the Hippocratic Corpus and other works.

Biography

Illustration of the story of Hippocrates refusing the presents of the Achaemenid Emperor Artaxerxes, who was asking for his services. Painted by Girodet, 1792.

Historians agree that Hippocrates was born around the year 460 BC on the Greek island of Kos; other biographical information, however, is likely to be untrue.

Soranus of Ephesus, a 2nd-century Greek physician, was Hippocrates' first biographer and is the source of most personal information about him. Later biographies are in the Suda of the 10th century AD, and in the works of John Tzetzes, which date from the 12th century AD. Hippocrates is mentioned in passing in the writings of two contemporaries: Plato, in Protagoras and Phaedrus, and Aristotle's Politics, which date from the 4th century BC.

Soranus wrote that Hippocrates' father was Heraclides, a physician, and his mother was Praxitela, daughter of Tizane. The two sons of Hippocrates, Thessalus and Draco, and his son-in-law, Polybus, were his students. According to Galen, a later physician, Polybus, was Hippocrates' true successor, while Thessalus and Draco each had a son named Hippocrates (Hippocrates III and IV).

Soranus said that Hippocrates learned medicine from his father and grandfather (Hippocrates I), and studied other subjects with Democritus and Gorgias. Hippocrates was probably trained at the asklepieion of Kos, and took lessons from the Thracian physician Herodicus of Selymbria. Plato mentions Hippocrates in two of his dialogues: in Protagoras, Plato describes Hippocrates as "Hippocrates of Kos, the Asclepiad"; while in Phaedrus, Plato suggests that "Hippocrates the Asclepiad" thought that a complete knowledge of the nature of the body was necessary for medicine. Hippocrates taught and practiced medicine throughout his life, traveling at least as far as Thessaly, Thrace, and the Sea of Marmara. Several different accounts of his death exist. He died, probably in Larissa, at the age of 83, 85 or 90, though some say he lived to be well over 100.

Hippocratic theory

It is thus with regard to the disease called Sacred: it appears to me to be nowise more divine nor more sacred than other diseases, but has a natural cause from the originates like other affections. Men regard its nature and cause as divine from ignorance and wonder....

— Hippocrates, On the Sacred Disease

Hippocrates is credited with being the first person to believe that diseases were caused naturally, not because of superstition and gods. Hippocrates was credited by the disciples of Pythagoras of allying philosophy and medicine. He separated the discipline of medicine from religion, believing and arguing that disease was not a punishment inflicted by the gods but rather the product of environmental factors, diet, and living habits. Indeed there is not a single mention of a mystical illness in the entirety of the Hippocratic Corpus. However, Hippocrates did work with many convictions that were based on what is now known to be incorrect anatomy and physiology, such as Humorism.

Ancient Greek schools of medicine were split (into the Knidian and Koan) on how to deal with disease. The Knidian school of medicine focused on diagnosis. Medicine at the time of Hippocrates knew almost nothing of human anatomy and physiology because of the Greek taboo forbidding the dissection of humans. The Knidian school consequently failed to distinguish when one disease caused many possible series of symptoms. The Hippocratic school or Koan school achieved greater success by applying general diagnoses and passive treatments. Its focus was on patient care and prognosis, not diagnosis. It could effectively treat diseases and allowed for a great development in clinical practice.

Hippocratic medicine and its philosophy are far removed from that of modern medicine. Now, the physician focuses on specific diagnosis and specialized treatment, both of which were espoused by the Knidian school. This shift in medical thought since Hippocrates' day has caused serious criticism over their denunciations; for example, the French doctor M. S. Houdart called the Hippocratic treatment a "meditation upon death".

Analogies have been drawn between Thucydides' historical method and the Hippocratic method, in particular the notion of "human nature" as a way of explaining foreseeable repetitions for future usefulness, for other times or for other cases.

Crisis

Asklepieion on Kos

An important concept in Hippocratic medicine was that of a crisis, a point in the progression of disease at which either the illness would begin to triumph and the patient would succumb to death, or the opposite would occur and natural processes would make the patient recover. After a crisis, a relapse might follow, and then another deciding crisis. According to this doctrine, crises tend to occur on critical days, which were supposed to be a fixed time after the contraction of a disease. If a crisis occurred on a day far from a critical day, a relapse might be expected. Galen believed that this idea originated with Hippocrates, though it is possible that it predated him.

Illustration of a Hippocratic bench, date unknown

Hippocratic medicine was humble and passive. The therapeutic approach was based on "the healing power of nature" ("vis medicatrix naturae" in Latin). According to this doctrine, the body contains within itself the power to re-balance the four humours and heal itself (physis). Hippocratic therapy focused on simply easing this natural process. To this end, Hippocrates believed "rest and immobilization [were] of capital importance". In general, the Hippocratic medicine was very kind to the patient; treatment was gentle, and emphasized keeping the patient clean and sterile. For example, only clean water or wine were ever used on wounds, though "dry" treatment was preferable. Soothing balms were sometimes employed.

Hippocrates was reluctant to administer drugs and engage in specialized treatment that might prove to be wrongly chosen; generalized therapy followed a generalized diagnosis. Generalized treatments he prescribed include fasting and the consumption of a mix of honey and vinegar. Hippocrates once said that "to eat when you are sick, is to feed your sickness". However, potent drugs were used on certain occasions. This passive approach was very successful in treating relatively simple ailments such as broken bones which required traction to stretch the skeletal system and relieve pressure on the injured area. The Hippocratic bench and other devices were used to this end.

One of the strengths of Hippocratic medicine was its emphasis on prognosis. At Hippocrates' time, medicinal therapy was quite immature, and often the best thing that physicians could do was to evaluate an illness and predict its likely progression based upon data collected in detailed case histories.

Professionalism

A number of ancient Greek surgical tools. On the left is a trephine; on the right, a set of scalpels. Hippocratic medicine made good use of these tools.

Hippocratic medicine was notable for its strict professionalism, discipline, and rigorous practice. The Hippocratic work On the Physician recommends that physicians always be well-kempt, honest, calm, understanding, and serious. The Hippocratic physician paid careful attention to all aspects of his practice: he followed detailed specifications for, "lighting, personnel, instruments, positioning of the patient, and techniques of bandaging and splinting" in the ancient operating room. He even kept his fingernails to a precise length.

The Hippocratic School gave importance to the clinical doctrines of observation and documentation. These doctrines dictate that physicians record their findings and their medicinal methods in a very clear and objective manner, so that these records may be passed down and employed by other physicians. Hippocrates made careful, regular note of many symptoms including complexion, pulse, fever, pains, movement, and excretions. He is said to have measured a patient's pulse when taking a case history to discover whether the patient was lying. Hippocrates extended clinical observations into family history and environment. "To him medicine owes the art of clinical inspection and observation."

Direct contributions to medicine

Clubbing of fingers in a patient with Eisenmenger's syndrome; first described by Hippocrates, clubbing is also known as "Hippocratic fingers".

Hippocrates and his followers were first to describe many diseases and medical conditions. He is given credit for the first description of clubbing of the fingers, an important diagnostic sign in chronic lung disease, lung cancer and cyanotic heart disease. For this reason, clubbed fingers are sometimes referred to as "Hippocratic fingers". Hippocrates was also the first physician to describe Hippocratic face in Prognosis. Shakespeare famously alludes to this description when writing of Falstaff's death in Act II, Scene iii. of Henry V.

Hippocrates began to categorize illnesses as acute, chronic, endemic and epidemic, and use terms such as, "exacerbation, relapse, resolution, crisis, paroxysm, peak, and convalescence." Another of Hippocrates' major contributions may be found in his descriptions of the symptomatology, physical findings, surgical treatment and prognosis of thoracic empyema, i.e. suppuration of the lining of the chest cavity. His teachings remain relevant to present-day students of pulmonary medicine and surgery. Hippocrates was the first documented chest surgeon and his findings and techniques, while crude, such as the use of lead pipes to drain chest wall abscess, are still valid.

The Hippocratic school of medicine described well the ailments of the human rectum and the treatment thereof, despite the school's poor theory of medicine. Hemorrhoids, for instance, though believed to be caused by an excess of bile and phlegm, were treated by Hippocratic physicians in relatively advanced ways. Cautery and excision are described in the Hippocratic Corpus, in addition to the preferred methods: ligating the hemorrhoids and drying them with a hot iron. Other treatments such as applying various salves are suggested as well. Today, "treatment [for hemorrhoids] still includes burning, strangling, and excising." Also, some of the fundamental concepts of proctoscopy outlined in the Corpus are still in use. For example, the uses of the rectal speculum, a common medical device, are discussed in the Hippocratic Corpus. This constitutes the earliest recorded reference to endoscopy. Hippocrates often used lifestyle modifications such as diet and exercise to treat diseases such as diabetes, what is today called lifestyle medicine.

Two popular but likely misquoted attributions to Hippocrates are "Let food be your medicine, and medicine be your food" and "Walking is man's best medicine". Both appear to be misquotations, and their exact origins remain unknown.

In 2017, researchers claimed that, while conducting restorations on the Saint Catherine's Monastery in South Sinai, they found a manuscript which contains a medical recipe of Hippocrates. The manuscript also contains three recipes with pictures of herbs that were created by an anonymous scribe.

Hippocratic Corpus

Main article: Hippocratic Corpus

 

A 12th-century Byzantine manuscript of the Oath in the form of a cross

The Hippocratic Corpus (Latin: Corpus Hippocraticum) is a collection of around seventy early medical works collected in Alexandrian Greece. It is written in Ionic Greek. The question of whether Hippocrates himself was the author of any of the treatises in the corpus has not been conclusively answered, but current debate revolves around only a few of the treatises seen as potentially authored by him. Because of the variety of subjects, writing styles and apparent date of construction, the Hippocratic Corpus could not have been written by one person (Ermerins numbers the authors at nineteen). The corpus came to be known by his name because of his fame, possibly all medical works were classified under 'Hippocrates' by a librarian in Alexandria. The volumes were probably produced by his students and followers.

The Hippocratic Corpus contains textbooks, lectures, research, notes and philosophical essays on various subjects in medicine, in no particular order. These works were written for different audiences, both specialists and laymen, and were sometimes written from opposing viewpoints; significant contradictions can be found between works in the Corpus. Notable among the treatises of the Corpus are The Hippocratic Oath; The Book of Prognostics; On Regimen in Acute Diseases; Aphorisms; On Airs, Waters and Places; Instruments of Reduction; On The Sacred Disease; etc.

Hippocratic Oath

Main article: Hippocratic Oath

The Hippocratic Oath, a seminal document on the ethics of medical practice, was attributed to Hippocrates in antiquity although new information shows it may have been written after his death. This is probably the most famous document of the Hippocratic Corpus. Recently the authenticity of the document's author has come under scrutiny. While the Oath is rarely used in its original form today, it serves as a foundation for other, similar oaths and laws that define good medical practice and morals. Such derivatives are regularly taken today by medical graduates about to enter medical practice.

Legacy

Mural painting showing Galen and Hippocrates. 12th century; Anagni, Italy

Although Hippocrates neither founded the school of medicine named after him, nor wrote most of the treatises attributed to him, he is traditionally regarded as the "Father of Medicine". His contributions revolutionized the practice of medicine; but after his death the advancement stalled. So revered was Hippocrates that his teachings were largely taken as too great to be improved upon and no significant advancements of his methods were made for a long time. The centuries after Hippocrates' death were marked as much by retrograde movement as by further advancement. For instance, "after the Hippocratic period, the practice of taking clinical case-histories died out," according to Fielding Garrison.

After Hippocrates, another significant physician was Galen, a Greek who lived from AD 129 to AD 200. Galen perpetuated the tradition of Hippocratic medicine, making some advancements, but also some regressions. In the Middle Ages, the Islamic world adopted Hippocratic methods and developed new medical technologies. After the European Renaissance, Hippocratic methods were revived in western Europe and even further expanded in the 19th century. Notable among those who employed Hippocrates' rigorous clinical techniques were Thomas Sydenham, William Heberden, Jean-Martin Charcot and William Osler. Henri Huchard, a French physician, said that these revivals make up "the whole history of internal medicine."

Image

Engraving: bust of Hippocrates by Paulus Pontius after Peter Paul Rubens, 1638

According to Aristotle's testimony, Hippocrates was known as "The Great Hippocrates". Concerning his disposition, Hippocrates was first portrayed as a "kind, dignified, old country doctor" and later as "stern and forbidding". He is certainly considered wise, of very great intellect and especially as very practical. Francis Adams describes him as "strictly the physician of experience and common sense."

His image as the wise, old doctor is reinforced by busts of him, which wear large beards on a wrinkled face. Many physicians of the time wore their hair in the style of Jove and Asklepius. Accordingly, the busts of Hippocrates that have been found could be only altered versions of portraits of these deities. Hippocrates and the beliefs that he embodied are considered medical ideals. Fielding Garrison, an authority on medical history, stated, "He is, above all, the exemplar of that flexible, critical, well-poised attitude of mind, ever on the lookout for sources of error, which is the very essence of the scientific spirit." "His figure... stands for all time as that of the ideal physician," according to A Short History of Medicine, inspiring the medical profession since his death.

Legends

The Travels of Sir John Mandeville reports (incorrectly) that Hippocrates was the ruler of the islands of "Kos and Lango" [sic], and recounts a legend about Hippocrates' daughter. She was transformed into a hundred-foot long dragon by the goddess Diana, and is the "lady of the manor" of an old castle. She emerges three times a year, and will be turned back into a woman if a knight kisses her, making the knight into her consort and ruler of the islands. Various knights try, but flee when they see the hideous dragon; they die soon thereafter. This is a version of the legend of Melusine.

Namesakes

Statue of Hippocrates in front of the Mayne Medical School in Brisbane

Some clinical symptoms and signs have been named after Hippocrates as he is believed to be the first person to describe those. Hippocratic face is the change produced in the countenance by death, or long sickness, excessive evacuations, excessive hunger, and the like. Clubbing, a deformity of the fingers and fingernails, is also known as Hippocratic fingers. Hippocratic succussion is the internal splashing noise of hydropneumothorax or pyopneumothorax. Hippocratic bench (a device which uses tension to aid in setting bones) and Hippocratic cap-shaped bandage are two devices named after Hippocrates. Hippocratic Corpus and Hippocratic Oath are also his namesakes. Risus sardonicus, a sustained spasming of the face muscles may also be termed the Hippocratic Smile. The most severe form of hair loss and baldness is called the Hippocratic form.

In the modern age, a lunar crater has been named Hippocrates. The Hippocratic Museum, a museum on the Greek island of Kos is dedicated to him. The Hippocrates Project is a program of the New York University Medical Center to enhance education through use of technology. Project Hippocrates (an acronym of "HIgh PerfOrmance Computing for Robot-AssisTEd Surgery") is an effort of the Carnegie Mellon School of Computer Science and Shadyside Medical Center, "to develop advanced planning, simulation, and execution technologies for the next generation of computer-assisted surgical robots." Both the Canadian Hippocratic Registry and American Hippocratic Registry are organizations of physicians who uphold the principles of the original Hippocratic Oath as inviolable through changing social times.

Genealogy

Hippocrates' legendary genealogy traces his paternal heritage directly to Asklepius and his maternal ancestry to Heracles. According to Tzetzes's Chiliades, the ahnentafel of Hippocrates II is:

A mosaic of Hippocrates on the floor of the Asclepieion of Kos, with Asklepius in the middle, 2nd–3rd century

1. Hippocrates II.
2. Heraclides
4. Hippocrates I.
8. Gnosidicus
16. Nebrus
32. Sostratus III.
64. Theodorus II.
128. Sostratus, II.
256. Thedorus
512. Cleomyttades
1024. Crisamis
2048. Dardanus
4096. Sostratus
8192. Hippolochus
16384. Podalirius
32768. Asklepius

Ecosystem-based adaptation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Ecosystem-based_adaptation 

Ecosystem-based adaptation (EbA) encompasses a broad set of approaches to adapt to climate change. They all involve the management of ecosystems and their services to reduce the vulnerability of human communities to the impacts of climate change. The Convention on Biological Diversity defines EbA as "the use of biodiversity and ecosystem services as part of an overall adaptation strategy to help people to adapt to the adverse effects of climate change".

EbA involves the conservation, sustainable management and restoration of ecosystems, such as forests, grasslands, wetlands, mangroves or coral reefs to reduce the harmful impacts of climate hazards including shifting patterns or levels of rainfall, changes in maximum and minimum temperatures, stronger storms, and increasingly variable climatic conditions. EbA measures can be implemented on their own or in combination with engineered approaches (such as the construction of water reservoirs or dykes), hybrid measures (such as artificial reefs) and approaches that strengthen the capacities of individuals and institutions to address climate risks (such as the introduction of early warning systems).

EbA is nested within the broader concept of nature-based solutions and complements and shares common elements with a wide variety of other approaches to building the resilience of social-ecological systems. These approaches include community-based adaptation, ecosystem-based disaster risk reduction, climate-smart agriculture, and green infrastructure, and often place emphasis on using participatory and inclusive processes and community/stakeholder engagement. The concept of EbA has been promoted through international fora, including the processes of the United Nations Framework Convention on Climate Change (UNFCCC) and the Convention on Biological Diversity (CBD). A number of countries make explicit references to EbA in their strategies for adaptation to climate change and their Nationally Determined Contributions (NDCs) under the Paris Agreement.

While the barriers to widespread uptake of EbA by public and private sector stakeholders and decision makers are substantial, cooperation toward generating a greater understanding of the potential of EbA is well established among researchers, advocates, and practitioners from nature conservation and sustainable development groups. EbA is increasingly viewed as an effective means of addressing the linked challenges of climate change and poverty in developing countries, where many people are dependent on natural resources for their lives and livelihoods.

Overview

Ecosystem-based Adaptation (EbA) describes a variety of approaches for adapting to climate change, all of which involve the management of ecosystems to reduce the vulnerability of human communities to the impacts of climate change such as storm and flood damage to physical assets, coastal erosion, salinisation of freshwater resources, and loss of agricultural productivity. EbA lies at the intersection of climate change adaptation, socio-economic development, and biodiversity conservation (see Figure 1).

While ecosystem services have always been used by societies, the term Ecosystem-based Adaptation was coined in 2008 by the International Union for Conservation of Nature (IUCN) and its member institutions at the UN Climate Change Convention Conference in 2008. EbA was officially defined in 2009 at the UN Convention on Biological Diversity Conference.

Adaptation to climate change hazards

Healthy ecosystems provide important ecosystem services that can contribute to climate change adaptation. For example, healthy mangrove ecosystems provide protection from the impacts of climate change, often for some of the world's most vulnerable people, by absorbing wave energy and storm surges, adapting to rising sea levels, and stabilizing shorelines from erosion. EbA focuses on benefits that humans derive from biodiversity and ecosystem services and how these benefits can be used for managing risk to climate change impacts. Adaptation to climate change is particularly urgent in developing countries and many Small Island Developing States that are already experiencing some of the most severe impacts of climate change, have economies that are highly sensitive to disruptions, and that have lower adaptive capacity.

Making active use of biodiversity and ecosystem services

EbA can involve a wide range of ecosystem management activities that aim to reduce the vulnerability of people to climate change hazards (such as rising sea levels, changing rainfall patterns, and stronger storms) through using nature. For example, EbA measures include coastal habitat restoration in ecosystems such as coral reefs, mangrove forests, and marshes to protect communities and infrastructure from storm surges; agroforestry to increase resilience of crops to droughts or excessive rainfall; integrated water resource management to cope with consecutive dry days and change in rainfall patterns; and sustainable forest management interventions to stabilise slopes, prevent landslides, and regulate water flow to prevent flash flooding (see Table 1).

Co-benefits of EbA

By deploying EbA, proponents cite that many other benefits to people and nature are delivered simultaneously. These correlated benefits include improved human health, socioeconomic development, food security and water security, disaster risk reduction, carbon sequestration, and biodiversity conservation. For example, restoration of ecosystems such as forests and coastal wetlands can contribute to food security and enhance livelihoods through the collection of non-timber forest products, maintain watershed functionality, and sequester carbon to mitigate global warming. Restoration of mangrove ecosystems can help increase food and livelihood security by supporting fisheries, and reduce disaster risk by decreasing wave height and strength during hurricanes and storms.

Implementation and examples of EbA

Examples of EbA measures and outcomes

Particular ecosystems can provide a variety specific climate change adaptation benefits (or services). The most suitable EbA measures will depend on local context, the health of the ecosystem and the primary climate change hazard that needs to be addressed. The below table provides an overview of these factors, common EbA measures and intended outcomes.

Table 1. Examples of EbA measures and outcomes The table shows climate hazards and their potential impacts on people, as well as examples of corresponding EbA measures. Many of the same climate hazards affect different ecosystems and have similar impacts on people, as such, the table illustrates the overlap between impacts, EbA measures and adaptation outcomes. Adapted from the PANORAMA database

Climate change hazards	Potential impacts on people	EbA measures by ecosystem type	Expected outcomes
Erratic rainfall Floods Shift of seasons Temperature increases Drought Extreme heat	Higher flood risks for people and infrastructure; Decrease in agricultural (and livestock) production; Food insecurities; Economic losses and/or insecurities; Threats to human health and well-being; Higher risk of heat strokes Lack of water	Mountains and forests: Sustainable mountain wetland management Forest and pasture restoration Inland waters: Conservation of wetlands and peat lands River basin restoration Trans-boundary water governance and ecosystem restoration Agriculture and drylands: Ecosystem restoration and agroforestry Using trees to adapt to changing seasons Intercropping of adapted species Sustainable livestock management and pasture restoration Sustainable dryland management Urban areas: Green aeration corridors for cities Storm water management using green spaces River restoration in urban areas Green facades for buildings	Improved water regulation; Erosion prevention; Improved water storage capacity; Flood risk reduction; Improved water provisioning; Improved water storage capacity; Adaptation to higher temperatures; Heat wave buffering
Storm surges Cyclones Sea level rise Salinisation Coastal erosion	Higher flood risks for people and infrastructure; Higher storm and cyclone risk for people and infrastructure; Decrease in agricultural (and livestock) production; Food insecurities; Economic losses and/or insecurities; Threats to human health and well-being; Lack of potable water	Marine and coastal: Mangrove restoration and coastal protection Coastal realignment Sustainable fishing and mangrove rehabilitation Coastal reef restoration	Storm and cyclone reduction; Flood risk reduction; Improved water quality; Adaptation to higher temperatures

Principles and standards for implementing EbA

Since the evolution of the concept and practice of EbA, various principles and standards have been developed to guide best practices for implementation. The guidelines adopted by the CBD build on these efforts and include a set of principles to guide planning and implementation. The principles are broadly clustered into four themes:

1. Building resilience and enhancing adaptive capacity through EbA interventions;
2. Ensuring inclusivity and equity in planning and implementation;
3. Consideration of multiple spatial and temporal scales in the design of EbA interventions;
4. Improving the effectiveness and efficiency of EbA, for example, by incorporating adaptive management, identifying limitations and trade-offs, integrating the knowledge of indigenous peoples and local communities.

These principles are complemented by safeguards, which are social and environmental measures to avoid unintended consequences of EbA to people, ecosystems and biodiversity.

Standards have also been developed to help practitioners understand what interventions qualify as EbA, including the elements of helping people adapt to climate change, making active use of biodiversity and ecosystem services, and being part of an overall adaptation strategy.

Challenges to be addressed for greater adoption of EbA

Although interest in Ecosystem-based Adaptation has grown, and meta-analyses of case studies are demonstrating the efficacy and cost-effectiveness of EbA interventions, there are recognised challenges that should be addressed or considered to increase adoption of the approach. These include:

Potential limitations of ecosystem services under a changing climate. One challenge facing EbA is the identification of limits and thresholds beyond which EbA might not deliver adaptation benefits and the extent ecosystems can provide ecosystem services under a changing climate.

Difficulty in monitoring, evaluation, and establishing the evidence base for effective EbA. Confusion around what Ecosystem-based Adaptation means has led to an array of different methodologies used for assessments, and the lack of consistent and comparable quantitative measures of EbA success and failure makes it difficult to argue the case for EbA in socio-economic terms. EbA research has also relied heavily on Western scientific knowledge without due consideration of local and traditional knowledge. In addition, it can be difficult to implement a plan for monitoring and evaluation due to potentially long timescales required to observe the impacts of EbA.

Governance and institutional constraints. Because EbA is a multi-sectoral policy issue, the challenges of governing and planning are immense. This is due in part to the fact that EbA involves both the sectors that manage ecosystems and those that benefit from ecosystem services.

Economic and financial constraints. Broad macroeconomic considerations such as economic development, poverty, and access to financial capital to implement climate adaptation options are contributing factors to constraints impeding greater uptake of EbA. Public and multilateral funding for EbA projects thus far has been available through the International Climate Initiative of the German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety, the Global Environment Facility, the Green Climate Fund, the European Union, the Department for International Development of the Government of the United Kingdom, the Swedish International Development Cooperation Agency and the Danish International Development Agency, among other sources.

Social and cultural barriers. A clear factor constraining EbA is varying perceptions of risks and cultural preferences for particular types of management approaches such as cultural preferences for what a particular landscape should look like. Potential stakeholders can hold negative perceptions about particular types of EbA strategies.

Policy frameworks

Several international policy fora have acknowledged the multiple roles that ecosystems play in delivering services and addressing global challenges, including those related to climate change, natural disasters, sustainable development, and biodiversity conservation.

Climate change policy

The Paris Agreement explicitly recognises nature's role in helping people and societies address climate change, calling on all Parties to acknowledge "the importance of ensuring the integrity of all ecosystems, including oceans, and the protection of biodiversity, recognised by some cultures as Mother Earth"; its Articles include several references to ecosystems, natural resources and forests.

This notion has translated into high-level national intent, as revealed by comparative analyses of the Nationally Determined Contributions (NDCs) submitted to the UN Framework Convention on Climate Change (UNFCCC) by signatories of the Paris Agreement. The UNFCCC also established the national adaptation plan (NAP) process as a way to facilitate adaptation planning in least developed countries (LDCs) and other developing countries. Because of their lower level of development, climate change risks magnify development challenges for LDCs.

Disaster risk reduction policy

Measures and interventions applied as part of EbA are often closely linked or similar to those employed under ecosystem-based disaster risk reduction (Eco-DRR). The Sendai Framework for Disaster Risk Reduction acknowledges that in order to strengthen disaster risk governance and manage disaster risk and risk reduction at global and regional levels, it is important "to promote transboundary cooperation to enable policy and planning for the implementation of ecosystem-based approaches with regard to shared resources, such as within river basins and along coastlines, to build resilience and reduce disaster risk, including epidemic and displacement risk".

Sustainable development policy

The Sustainable Development Goals (SDGs) are a collection of 17 global goals set by the United Nations General Assembly in 2015. Biodiversity and ecosystems feature prominently across many of the SDGs and associated targets. They contribute directly to human well-being and development priorities. Biodiversity is at the centre of many economic activities, particularly those related to crop and livestock agriculture, forestry, and fisheries. Globally, nearly half of the human population is directly dependent on natural resources for its livelihood, and many of the most vulnerable people depend directly on biodiversity to fulfil their daily subsistence needs. Ecosystem-based Adaptation offers potential to contribute towards the implementation of numerous SDGs, including the goals related to climate adaptation (SDG 13), eliminating poverty and hunger (SDGs 1 and 2), ensuring livelihoods and economic growth (SDG 8) and life on land and life under water (SDGs 14 and 15), among others.

Biodiversity conservation policy

The Strategic Plan for Biodiversity 2011-2020 and the Aichi Biodiversity Targets, under the Convention on Biological Diversity (CBD), aim to halt the loss of biodiversity to ensure ecosystems are resilient and continue to provide essential services. Most recently, the Conference of the Parties has adopted voluntary guidelines for the design and effective implementation of ecosystem-based approaches to adaptation and disaster risk reduction.

EbA and similar approaches have been called for in other policy frameworks, including the United Nations Convention to Combat Desertification (UNCCD) and the Ramsar Convention.

EbA knowledge exchange platforms

The following is an alphabetical list of EbA networks, working groups, and platforms that are exchanging knowledge and experiences in an effort to address and overcome the challenges of implementing EbA. This selection is not exhaustive.

AdaptationCommunity

Coastal EbA

EbA Community

Friends of EbA

International EbA Community of Practice

PANORAMA Solutions – EbA Portal

We Adapt

Group (mathematics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Group_(mathematics) 

The manipulations of the Rubik's Cube form the Rubik's Cube group.

In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.

In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in geometry, and also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity. Point groups describe symmetry in molecular chemistry.

The concept of a group arose in the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term group (French: groupe) for the symmetry group of the roots of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.

Definition and illustration

First example: the integers

One of the more familiar groups is the set of integers

$${\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}$$

together with addition. For any two integers ${\displaystyle a}$ and ${\displaystyle b}$, the sum ${\displaystyle a+b}$ is also an integer; this closure property says that ${\displaystyle +}$ is a binary operation on ${\displaystyle \mathbb {Z} }$. The following properties of integer addition serve as a model for the group axioms in the definition below.

• For all integers ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$, one has ${\displaystyle (a+b)+c=a+(b+c)}$. Expressed in words, adding ${\displaystyle a}$ to ${\displaystyle b}$ first, and then adding the result to ${\displaystyle c}$ gives the same final result as adding ${\displaystyle a}$ to the sum of ${\displaystyle b}$ and ${\displaystyle c}$. This property is known as associativity.
• If ${\displaystyle a}$ is any integer, then ${\displaystyle 0+a=a}$ and ${\displaystyle a+0=a}$. Zero is called the identity element of addition because adding it to any integer returns the same integer.
• For every integer ${\displaystyle a}$, there is an integer ${\displaystyle b}$ such that ${\displaystyle a+b=0}$ and ${\displaystyle b+a=0}$. The integer ${\displaystyle b}$ is called the inverse element of the integer ${\displaystyle a}$ and is denoted ${\displaystyle -a}$.

The integers, together with the operation ${\displaystyle +}$, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

Definition

The axioms for a group are short and natural... Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.

Richard Borcherds in Mathematicians: An Outer View of the Inner World

A group is a set ${\displaystyle G}$ together with a binary operation on ${\displaystyle G}$, here denoted "${\displaystyle \cdot }$", that combines any two elements ${\displaystyle a}$ and ${\displaystyle b}$ to form an element of ${\displaystyle G}$, denoted ${\displaystyle a\cdot b}$, such that the following three requirements, known as group axioms, are satisfied:

Associativity

For all ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ in ${\displaystyle G}$, one has ${\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}$.

Identity element

There exists an element ${\displaystyle e}$ in ${\displaystyle G}$ such that, for every ${\displaystyle a}$ in ${\displaystyle G}$, one has ${\displaystyle e\cdot a=a}$ and ${\displaystyle a\cdot e=a}$.

Such an element is unique (see below). It is called the identity element of the group.

Inverse element

For each ${\displaystyle a}$ in ${\displaystyle G}$, there exists an element ${\displaystyle b}$ in ${\displaystyle G}$ such that ${\displaystyle a\cdot b=e}$ and ${\displaystyle b\cdot a=e}$, where ${\displaystyle e}$ is the identity element.

For each ${\displaystyle a}$, the element ${\displaystyle b}$ is unique (see below); it is called the inverse of ${\displaystyle a}$ and is commonly denoted ${\displaystyle a^{-1}}$.

Notation and terminology

Formally, the group is the ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the underlying set of the group, and the operation is called the group operation or the group law.

A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.

For example, consider the set of real numbers ${\displaystyle \mathbb {R} }$, which has the operations of addition ${\displaystyle a+b}$ and multiplication ${\displaystyle ab}$. Formally, ${\displaystyle \mathbb {R} }$ is a set, ${\displaystyle (\mathbb {R} ,+)}$ is a group, and ${\displaystyle (\mathbb {R} ,+,\cdot )}$ is a field. But it is common to write ${\displaystyle \mathbb {R} }$ to denote any of these three objects.

The additive group of the field ${\displaystyle \mathbb {R} }$ is the group whose underlying set is ${\displaystyle \mathbb {R} }$ and whose operation is addition. The multiplicative group of the field ${\displaystyle \mathbb {R} }$ is the group ${\displaystyle \mathbb {R} ^{\times }}$ whose underlying set is the set of nonzero real numbers ${\displaystyle \mathbb {R} \smallsetminus \{0\}}$ and whose operation is multiplication.

More generally, one speaks of an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted ${\displaystyle 0}$, and the inverse of an element ${\displaystyle x}$ is denoted ${\displaystyle -x}$. Similarly, one speaks of a multiplicative group whenever the group operation is notated as multiplication; in this case, the identity is typically denoted ${\displaystyle 1}$, and the inverse of an element ${\displaystyle x}$ is denoted ${\displaystyle x^{-1}}$. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, ${\displaystyle ab}$ instead of ${\displaystyle a\cdot b}$.

The definition of a group does not require that ${\displaystyle a\cdot b=b\cdot a}$ for all elements ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle G}$. If this additional condition holds, then the operation is said to be commutative, and the group is called an abelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.

Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions, the operation is often function composition ${\displaystyle f\circ g}$; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups, and automorphism groups, the symbol ${\displaystyle \circ }$ is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.

Second example: a symmetry group

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries. A square has eight symmetries. These are:

The elements of the symmetry group of the square, ${\displaystyle \mathrm {D} _{4}}$. Vertices are identified by color or number.

${\displaystyle \mathrm {id} }$ (keeping it as it is)	${\displaystyle r_{1}}$ (rotation by 90° clockwise)	${\displaystyle r_{2}}$ (rotation by 180°)	${\displaystyle r_{3}}$ (rotation by 270° clockwise)
${\displaystyle f_{\mathrm {v} }}$ (vertical reflection)	${\displaystyle f_{\mathrm {h} }}$ (horizontal reflection)	${\displaystyle f_{\mathrm {d} }}$ (diagonal reflection)	${\displaystyle f_{\mathrm {c} }}$ (counter-diagonal reflection)

• the identity operation leaving everything unchanged, denoted id;
• rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by ${\displaystyle r_{1}}$, ${\displaystyle r_{2}}$ and ${\displaystyle r_{3}}$, respectively;
• reflections about the horizontal and vertical middle line (${\displaystyle f_{\mathrm {v} }}$ and ${\displaystyle f_{\mathrm {h} }}$), or through the two diagonals (${\displaystyle f_{\mathrm {d} }}$ and ${\displaystyle f_{\mathrm {c} }}$).

These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, ${\displaystyle r_{1}}$ sends a point to its rotation 90° clockwise around the square's center, and ${\displaystyle f_{\mathrm {h} }}$ sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the dihedral group of degree four, denoted ${\displaystyle \mathrm {D} _{4}}$. The underlying set of the group is the above set of symmetries, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first ${\displaystyle a}$ and then ${\displaystyle b}$ is written symbolically from right to left as ${\displaystyle b\circ a}$ ("apply the symmetry ${\displaystyle b}$ after performing the symmetry ${\displaystyle a}$"). This is the usual notation for composition of functions.

The group table lists the results of all such compositions possible. For example, rotating by 270° clockwise (${\displaystyle r_{3}}$) and then reflecting horizontally (${\displaystyle f_{\mathrm {h} }}$) is the same as performing a reflection along the diagonal (${\displaystyle f_{\mathrm {d} }}$). Using the above symbols, highlighted in blue in the group table:

$${\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}$$

Group table of ${\displaystyle \mathrm {D} _{4}}$

${\displaystyle \circ }$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {c} }}$
${\displaystyle \mathrm {id} }$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {c} }}$
${\displaystyle r_{1}}$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle \mathrm {id} }$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {h} }}$
${\displaystyle r_{2}}$	${\displaystyle r_{2}}$	${\displaystyle r_{3}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {d} }}$
${\displaystyle r_{3}}$	${\displaystyle r_{3}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{1}}$	${\displaystyle r_{2}}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {v} }}$
${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{2}}$	${\displaystyle r_{1}}$	${\displaystyle r_{3}}$
${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle r_{2}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{3}}$	${\displaystyle r_{1}}$
${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle r_{3}}$	${\displaystyle r_{1}}$	${\displaystyle \mathrm {id} }$	${\displaystyle r_{2}}$
${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {c} }}$	${\displaystyle f_{\mathrm {v} }}$	${\displaystyle f_{\mathrm {d} }}$	${\displaystyle f_{\mathrm {h} }}$	${\displaystyle r_{1}}$	${\displaystyle r_{3}}$	${\displaystyle r_{2}}$	${\displaystyle \mathrm {id} }$
The elements ${\displaystyle \mathrm {id} }$, ${\displaystyle r_{1}}$, ${\displaystyle r_{2}}$, and ${\displaystyle r_{3}}$ form a subgroup whose group table is highlighted in   red (upper left region). A left and right coset of this subgroup are highlighted in   green (in the last row) and   yellow (last column), respectively. The result of the composition ${\displaystyle f_{\mathrm {h} }\circ r_{3}}$, the symmetry ${\displaystyle f_{\mathrm {d} }}$, is highlighted in   blue (below table center).

Given this set of symmetries and the described operation, the group axioms can be understood as follows.

Binary operation: Composition is a binary operation. That is, ${\displaystyle a\circ b}$ is a symmetry for any two symmetries ${\displaystyle a}$ and ${\displaystyle b}$. For example,

$${\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}$$

that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (${\displaystyle f_{\mathrm {c} }}$). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the group table.

Associativity: The associativity axiom deals with composing more than two symmetries: Starting with three elements ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ of ${\displaystyle \mathrm {D} _{4}}$, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose ${\displaystyle a}$ and ${\displaystyle b}$ into a single symmetry, then to compose that symmetry with ${\displaystyle c}$. The other way is to first compose ${\displaystyle b}$ and ${\displaystyle c}$, then to compose the resulting symmetry with ${\displaystyle a}$. These two ways must give always the same result, that is,

$${\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}$$

For example, ${\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}$ can be checked using the group table:

$${\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}$$

Identity element: The identity element is ${\displaystyle \mathrm {id} }$, as it does not change any symmetry ${\displaystyle a}$ when composed with it either on the left or on the right.

Inverse element: Each symmetry has an inverse: ${\displaystyle \mathrm {id} }$, the reflections ${\displaystyle f_{\mathrm {h} }}$, ${\displaystyle f_{\mathrm {v} }}$, ${\displaystyle f_{\mathrm {d} }}$, ${\displaystyle f_{\mathrm {c} }}$ and the 180° rotation ${\displaystyle r_{2}}$ are their own inverse, because performing them twice brings the square back to its original orientation. The rotations ${\displaystyle r_{3}}$ and ${\displaystyle r_{1}}$ are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.

In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in ${\displaystyle \mathrm {D} _{4}}$, as, for example, ${\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}$ but ${\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}$. In other words, ${\displaystyle \mathrm {D} _{4}}$ is not abelian.

History

Main article: History of group theory

The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation ${\displaystyle \theta ^{n}=1}$ (1854) gives the first abstract definition of a finite group.

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.

The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss's number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers.

The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.

The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the examples below illustrate.

Elementary consequences of the group axioms

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. For example, repeated applications of the associativity axiom show that the unambiguity of

$${\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}$$

generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.

Individual axioms may be "weakened" to assert only the existence of a left identity and left inverses. From these one-sided axioms, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are no weaker.

Uniqueness of identity element

The group axioms imply that the identity element is unique: If ${\displaystyle e}$ and ${\displaystyle f}$ are identity elements of a group, then ${\displaystyle e=e\cdot f=f}$. Therefore, it is customary to speak of the identity.

Uniqueness of inverses

The group axioms also imply that the inverse of each element is unique: If a group element ${\displaystyle a}$ has both ${\displaystyle b}$ and ${\displaystyle c}$ as inverses, then

${\displaystyle b}$	${\displaystyle {}={}}$	${\displaystyle b\cdot e}$		since ${\displaystyle e}$ is the identity element
	${\displaystyle {}={}}$	${\displaystyle b\cdot (a\cdot c)}$		since ${\displaystyle c}$ is an inverse of ${\displaystyle a}$, so ${\displaystyle e=a\cdot c}$
	${\displaystyle {}={}}$	${\displaystyle (b\cdot a)\cdot c}$		by associativity, which allows rearranging the parentheses
	${\displaystyle {}={}}$	${\displaystyle e\cdot c}$		since ${\displaystyle b}$ is an inverse of ${\displaystyle a}$, so ${\displaystyle b\cdot a=e}$
	${\displaystyle {}={}}$	${\displaystyle c}$		since ${\displaystyle e}$ is the identity element.

Therefore, it is customary to speak of the inverse of an element.

Division

Given elements ${\displaystyle a}$ and ${\displaystyle b}$ of a group ${\displaystyle G}$, there is a unique solution ${\displaystyle x}$ in ${\displaystyle G}$ to the equation ${\displaystyle a\cdot x=b}$, namely ${\displaystyle a^{-1}\cdot b}$. (One usually avoids using fraction notation ${\displaystyle {\tfrac {b}{a}}}$ unless ${\displaystyle G}$ is abelian, because of the ambiguity of whether it means ${\displaystyle a^{-1}\cdot b}$ or ${\displaystyle b\cdot a^{-1}}$.) It follows that for each ${\displaystyle a}$ in ${\displaystyle G}$, the function ${\displaystyle G\to G}$ that maps each ${\displaystyle x}$ to ${\displaystyle a\cdot x}$ is a bijection; it is called left multiplication by ${\displaystyle a}$ or left translation by ${\displaystyle a}$.

Similarly, given ${\displaystyle a}$ and ${\displaystyle b}$, the unique solution to ${\displaystyle x\cdot a=b}$ is ${\displaystyle b\cdot a^{-1}}$. For each ${\displaystyle a}$, the function ${\displaystyle G\to G}$ that maps each ${\displaystyle x}$ to ${\displaystyle x\cdot a}$ is a bijection called right multiplication by ${\displaystyle a}$ or right translation by ${\displaystyle a}$.

Basic concepts

The following sections use mathematical symbols such as ${\displaystyle X=\{x,y,z\}}$ to denote a set ${\displaystyle X}$ containing elements ${\displaystyle x,y,}$ and ${\displaystyle z}$, or ${\displaystyle x\in X}$ to state that ${\displaystyle x}$ is an element of ${\displaystyle X.}$ The notation ${\displaystyle f:X\to Y}$ means ${\displaystyle f}$ is a function associating to every element of ${\displaystyle X}$ an element of ${\displaystyle Y.}$

When studying sets, one uses concepts such as subset, function, and quotient by an equivalence relation. When studying groups, one uses instead subgroups, homomorphisms, and quotient groups. These are the analogues that take the group structure into account.

Group homomorphisms

Main article: Group homomorphism

Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A homomorphism from a group ${\displaystyle (G,\cdot )}$ to a group ${\displaystyle (H,*)}$ is a function ${\displaystyle \varphi :G\to H}$ such that

${\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)}$ for all elements ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle G}$.

It would be natural to require also that ${\displaystyle \varphi }$ respect identities, ${\displaystyle \varphi (1_{G})=1_{H}}$, and inverses, ${\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}$ for all ${\displaystyle a}$ in ${\displaystyle G}$. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.

The identity homomorphism of a group ${\displaystyle G}$ is the homomorphism ${\displaystyle \iota _{G}:G\to G}$ that maps each element of ${\displaystyle G}$ to itself. An inverse homomorphism of a homomorphism ${\displaystyle \varphi :G\to H}$ is a homomorphism ${\displaystyle \psi :H\to G}$ such that ${\displaystyle \psi \circ \varphi =\iota _{G}}$ and ${\displaystyle \varphi \circ \psi =\iota _{H}}$, that is, such that ${\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}$ for all ${\displaystyle g}$ in ${\displaystyle G}$ and such that ${\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}$ for all ${\displaystyle h}$ in ${\displaystyle H}$. An isomorphism is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism. Groups ${\displaystyle G}$ and ${\displaystyle H}$ are called isomorphic if there exists an isomorphism ${\displaystyle \varphi :G\to H}$. In this case, ${\displaystyle H}$ can be obtained from ${\displaystyle G}$ simply by renaming its elements according to the function ${\displaystyle \varphi }$; then any statement true for ${\displaystyle G}$ is true for ${\displaystyle H}$, provided that any specific elements mentioned in the statement are also renamed.

The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.

Subgroups

Main article: Subgroup

Informally, a subgroup is a group ${\displaystyle H}$ contained within a bigger one, ${\displaystyle G}$: it has a subset of the elements of ${\displaystyle G}$, with the same operation. Concretely, this means that the identity element of ${\displaystyle G}$ must be contained in ${\displaystyle H}$, and whenever ${\displaystyle h_{1}}$ and ${\displaystyle h_{2}}$ are both in ${\displaystyle H}$, then so are ${\displaystyle h_{1}\cdot h_{2}}$ and ${\displaystyle h_{1}^{-1}}$, so the elements of ${\displaystyle H}$, equipped with the group operation on ${\displaystyle G}$ restricted to ${\displaystyle H}$, indeed form a group. In this case, the inclusion map ${\displaystyle H\to G}$ is a homomorphism.

In the example of symmetries of a square, the identity and the rotations constitute a subgroup ${\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}$, highlighted in red in the group table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The subgroup test provides a necessary and sufficient condition for a nonempty subset H of a group G to be a subgroup: it is sufficient to check that ${\displaystyle g^{-1}\cdot h\in H}$ for all elements ${\displaystyle g}$ and ${\displaystyle h}$ in ${\displaystyle H}$. Knowing a group's subgroups is important in understanding the group as a whole.

Given any subset ${\displaystyle S}$ of a group ${\displaystyle G}$, the subgroup generated by ${\displaystyle S}$ consists of all products of elements of ${\displaystyle S}$ and their inverses. It is the smallest subgroup of ${\displaystyle G}$ containing ${\displaystyle S}$. In the example of symmetries of a square, the subgroup generated by ${\displaystyle r_{2}}$ and ${\displaystyle f_{\mathrm {v} }}$ consists of these two elements, the identity element ${\displaystyle \mathrm {id} }$, and the element ${\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}$. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.

An injective homomorphism ${\displaystyle \phi \colon G'\to G}$ factors canonically as an isomorphism followed by an inclusion, ${\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}$ for some subgroup H of G. Injective homomorphisms are the monomorphisms in the category of groups.

Cosets

Main article: Coset

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup ${\displaystyle H}$ determines left and right cosets, which can be thought of as translations of ${\displaystyle H}$ by an arbitrary group element ${\displaystyle g}$. In symbolic terms, the left and right cosets of ${\displaystyle H}$, containing an element ${\displaystyle g}$, are

${\displaystyle gH=\{g\cdot h\mid h\in H\}}$ and ${\displaystyle Hg=\{h\cdot g\mid h\in H\}}$, respectively.

The left cosets of any subgroup ${\displaystyle H}$ form a partition of ${\displaystyle G}$; that is, the union of all left cosets is equal to ${\displaystyle G}$ and two left cosets are either equal or have an empty intersection. The first case ${\displaystyle g_{1}H=g_{2}H}$ happens precisely when ${\displaystyle g_{1}^{-1}\cdot g_{2}\in H}$, i.e., when the two elements differ by an element of ${\displaystyle H}$. Similar considerations apply to the right cosets of ${\displaystyle H}$. The left cosets of ${\displaystyle H}$ may or may not be the same as its right cosets. If they are (that is, if all ${\displaystyle g}$ in ${\displaystyle G}$ satisfy ${\displaystyle gH=Hg}$), then ${\displaystyle H}$ is said to be a normal subgroup.

In ${\displaystyle \mathrm {D} _{4}}$, the group of symmetries of a square, with its subgroup ${\displaystyle R}$ of rotations, the left cosets ${\displaystyle gR}$ are either equal to ${\displaystyle R}$, if ${\displaystyle g}$ is an element of ${\displaystyle R}$ itself, or otherwise equal to ${\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}$ (highlighted in green in the group table of ${\displaystyle \mathrm {D} _{4}}$). The subgroup ${\displaystyle R}$ is normal, because ${\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}$ and similarly for the other elements of the group. (In fact, in the case of ${\displaystyle \mathrm {D} _{4}}$, the cosets generated by reflections are all equal: ${\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}$.)

Quotient groups

Main article: Quotient group

Suppose that ${\displaystyle N}$ is a normal subgroup of a group ${\displaystyle G}$, and

$${\displaystyle G/N=\{gN\mid g\in G\}}$$

denotes its set of cosets. Then there is a unique group law on ${\displaystyle G/N}$ for which the map ${\displaystyle G\to G/N}$ sending each element ${\displaystyle g}$ to ${\displaystyle gN}$ is a homomorphism. Explicitly, the product of two cosets ${\displaystyle gN}$ and ${\displaystyle hN}$ is ${\displaystyle (gh)N}$, the coset ${\displaystyle eN=N}$ serves as the identity of ${\displaystyle G/N}$, and the inverse of ${\displaystyle gN}$ in the quotient group is ${\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}$. The group ${\displaystyle G/N}$, read as "${\displaystyle G}$ modulo ${\displaystyle N}$", is called a quotient group or factor group. The quotient group can alternatively be characterized by a universal property.

Group table of the quotient group ${\displaystyle \mathrm {D} _{4}/R}$

${\displaystyle \cdot }$	${\displaystyle R}$	${\displaystyle U}$
${\displaystyle R}$	${\displaystyle R}$	${\displaystyle U}$
${\displaystyle U}$	${\displaystyle U}$	${\displaystyle R}$

The elements of the quotient group ${\displaystyle \mathrm {D} _{4}/R}$ are ${\displaystyle R}$ and ${\displaystyle U=f_{\mathrm {v} }R}$. The group operation on the quotient is shown in the table. For example, ${\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}$. Both the subgroup ${\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}$ and the quotient ${\displaystyle \mathrm {D} _{4}/R}$ are abelian, but ${\displaystyle \mathrm {D} _{4}}$ is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the semidirect product construction; ${\displaystyle \mathrm {D} _{4}}$ is an example.

The first isomorphism theorem implies that any surjective homomorphism ${\displaystyle \phi \colon G\to H}$ factors canonically as a quotient homomorphism followed by an isomorphism: ${\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}$. Surjective homomorphisms are the epimorphisms in the category of groups.

Presentations

Main article: Presentation of a group

Every group is isomorphic to a quotient of a free group, in many ways.

For example, the dihedral group ${\displaystyle \mathrm {D} _{4}}$ is generated by the right rotation ${\displaystyle r_{1}}$ and the reflection ${\displaystyle f_{\mathrm {v} }}$ in a vertical line (every element of ${\displaystyle \mathrm {D} _{4}}$ is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism φ from the free group ${\displaystyle \langle r,f\rangle }$ on two generators to ${\displaystyle \mathrm {D} _{4}}$ sending ${\displaystyle r}$ to ${\displaystyle r_{1}}$ and ${\displaystyle f}$ to ${\displaystyle f_{1}}$. Elements in ${\displaystyle \ker \phi }$ are called relations; examples include ${\displaystyle r^{4},r^{2},(r\cdot f)^{2}}$. In fact, it turns out that ${\displaystyle \ker \phi }$ is the smallest normal subgroup of ${\displaystyle \langle r,f\rangle }$ containing these three elements; in other words, all relations are consequences of these three. The quotient of the free group by this normal subgroup is denoted ${\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }$. This is called a presentation of ${\displaystyle \mathrm {D} _{4}}$ by generators and relations, because the first isomorphism theorem for φ yields an isomorphism ${\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}$.

A presentation of a group can be used to construct the Cayley graph, a graphical depiction of a discrete group.

Examples and applications

Main article: Examples of groups

 

A periodic wallpaper pattern gives rise to a wallpaper group.

 

The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers.

Examples and applications of groups abound. A starting point is the group ${\displaystyle \mathbb {Z} }$ of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.

Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups. For example, elements of the fundamental group are represented by loops. The second image shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.

In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory.

In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.

Numbers

Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.

Integers

The group of integers ${\displaystyle \mathbb {Z} }$ under addition, denoted ${\displaystyle \left(\mathbb {Z} ,+\right)}$, has been described above. The integers, with the operation of multiplication instead of addition, ${\displaystyle \left(\mathbb {Z} ,\cdot \right)}$ do not form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, ${\displaystyle a=2}$ is an integer, but the only solution to the equation ${\displaystyle a\cdot b=1}$ in this case is ${\displaystyle b={\tfrac {1}{2}}}$, which is a rational number, but not an integer. Hence not every element of ${\displaystyle \mathbb {Z} }$ has a (multiplicative) inverse.

Rationals

The desire for the existence of multiplicative inverses suggests considering fractions

$${\displaystyle {\frac {a}{b}}.}$$

Fractions of integers (with ${\displaystyle b}$ nonzero) are known as rational numbers. The set of all such irreducible fractions is commonly denoted ${\displaystyle \mathbb {Q} }$. There is still a minor obstacle for ${\displaystyle \left(\mathbb {Q} ,\cdot \right)}$, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no ${\displaystyle x}$ such that ${\displaystyle x\cdot 0=1}$), ${\displaystyle \left(\mathbb {Q} ,\cdot \right)}$ is still not a group.

However, the set of all nonzero rational numbers ${\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}$ does form an abelian group under multiplication, also denoted ${\displaystyle \mathbb {Q} ^{\times }}$. Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of ${\displaystyle a/b}$ is ${\displaystyle b/a}$, therefore the axiom of the inverse element is satisfied.

The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if division by other than zero is possible, such as in ${\displaystyle \mathbb {Q} }$ – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.

Modular arithmetic

Main article: Modular arithmetic

 

The hours on a clock form a group that uses addition modulo 12. Here, 9 + 4 ≡ 1.

Modular arithmetic for a modulus ${\displaystyle n}$ defines any two elements ${\displaystyle a}$ and ${\displaystyle b}$ that differ by a multiple of ${\displaystyle n}$ to be equivalent, denoted by ${\displaystyle a\equiv b{\pmod {n}}}$. Every integer is equivalent to one of the integers from ${\displaystyle 0}$ to ${\displaystyle n-1}$, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined in this way for the integers from ${\displaystyle 0}$ to ${\displaystyle n-1}$, forms a group, denoted as ${\displaystyle \mathrm {Z} _{n}}$ or ${\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}$, with ${\displaystyle 0}$ as the identity element and ${\displaystyle n-a}$ as the inverse element of ${\displaystyle a}$.

A familiar example is addition of hours on the face of a clock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on ${\displaystyle 9}$ and is advanced ${\displaystyle 4}$ hours, it ends up on ${\displaystyle 1}$, as shown in the illustration. This is expressed by saying that ${\displaystyle 9+4}$ is congruent to ${\displaystyle 1}$ "modulo ${\displaystyle 12}$" or, in symbols,

$${\displaystyle 9+4\equiv 1{\pmod {12}}.}$$

For any prime number ${\displaystyle p}$, there is also the multiplicative group of integers modulo ${\displaystyle p}$. Its elements can be represented by ${\displaystyle 1}$ to ${\displaystyle p-1}$. The group operation, multiplication modulo ${\displaystyle p}$, replaces the usual product by its representative, the remainder of division by ${\displaystyle p}$. For example, for ${\displaystyle p=5}$, the four group elements can be represented by ${\displaystyle 1,2,3,4}$. In this group, ${\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}$, because the usual product ${\displaystyle 16}$ is equivalent to ${\displaystyle 1}$: when divided by ${\displaystyle 5}$ it yields a remainder of ${\displaystyle 1}$. The primality of ${\displaystyle p}$ ensures that the usual product of two representatives is not divisible by ${\displaystyle p}$, and therefore that the modular product is nonzero. The identity element is represented by ${\displaystyle 1}$, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer ${\displaystyle a}$ not divisible by ${\displaystyle p}$, there exists an integer ${\displaystyle b}$ such that

$${\displaystyle a\cdot b\equiv 1{\pmod {p}},}$$

that is, such that ${\displaystyle p}$ evenly divides ${\displaystyle a\cdot b-1}$. The inverse ${\displaystyle b}$ can be found by using Bézout's identity and the fact that the greatest common divisor ${\displaystyle \gcd(a,p)}$ equals ${\displaystyle 1}$. In the case ${\displaystyle p=5}$ above, the inverse of the element represented by ${\displaystyle 4}$ is that represented by ${\displaystyle 4}$, and the inverse of the element represented by ${\displaystyle 3}$ is represented by ${\displaystyle 2}$, as ${\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}$. Hence all group axioms are fulfilled. This example is similar to ${\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}$ above: it consists of exactly those elements in the ring ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ that have a multiplicative inverse. These groups, denoted ${\displaystyle \mathbb {F} _{p}^{\times }}$, are crucial to public-key cryptography.

Cyclic groups

Main article: Cyclic group

 

The 6th complex roots of unity form a cyclic group. ${\displaystyle z}$ is a primitive element, but ${\displaystyle z^{2}}$ is not, because the odd powers of ${\displaystyle z}$ are not a power of ${\displaystyle z^{2}}$.

A cyclic group is a group all of whose elements are powers of a particular element ${\displaystyle a}$. In multiplicative notation, the elements of the group are

$${\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}$$

where ${\displaystyle a^{2}}$ means ${\displaystyle a\cdot a}$, ${\displaystyle a^{-3}}$ stands for ${\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}$, etc. Such an element ${\displaystyle a}$ is called a generator or a primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as

$${\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}$$

In the groups ${\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}$ introduced above, the element ${\displaystyle 1}$ is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are ${\displaystyle 1}$. Any cyclic group with ${\displaystyle n}$ elements is isomorphic to this group. A second example for cyclic groups is the group of ${\displaystyle n}$th complex roots of unity, given by complex numbers ${\displaystyle z}$ satisfying ${\displaystyle z^{n}=1}$. These numbers can be visualized as the vertices on a regular ${\displaystyle n}$-gon, as shown in blue in the image for ${\displaystyle n=6}$. The group operation is multiplication of complex numbers. In the picture, multiplying with ${\displaystyle z}$ corresponds to a counter-clockwise rotation by 60°. From field theory, the group ${\displaystyle \mathbb {F} _{p}^{\times }}$ is cyclic for prime ${\displaystyle p}$: for example, if ${\displaystyle p=5}$, ${\displaystyle 3}$ is a generator since ${\displaystyle 3^{1}=3}$, ${\displaystyle 3^{2}=9\equiv 4}$, ${\displaystyle 3^{3}\equiv 2}$, and ${\displaystyle 3^{4}\equiv 1}$.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element ${\displaystyle a}$, all the powers of ${\displaystyle a}$ are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to ${\displaystyle (\mathbb {Z} ,+)}$, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.

The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.

Symmetry groups

Main article: Symmetry group

See also: Molecular symmetry, Space group, Point group, and Symmetry in physics

 

The (2,3,7) triangle group, a hyperbolic reflection group, acts on this tiling of the hyperbolic plane

Symmetry groups are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element can be associated to some operation on X and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on.

In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.

Group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.

Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.

Buckminsterfullerene displays icosahedral symmetry	Ammonia, NH3. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.	Cubane C8H8 features octahedral symmetry.	The tetrachloroplatinate(II) ion, [PtCl4]2- exhibits square-planar geometry

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.

General linear group and representation theory

Main articles: General linear group, Representation theory, and Character theory

 

Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the ${\displaystyle x}$-coordinate by factor 2.

Matrix groups consist of matrices together with matrix multiplication. The general linear group ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$ consists of all invertible ${\displaystyle n}$-by-${\displaystyle n}$ matrices with real entries. Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group ${\displaystyle \mathrm {SO} (n)}$. It describes all possible rotations in ${\displaystyle n}$ dimensions. Rotation matrices in this group are used in computer graphics.

Representation theory is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of group representations are linear representations in which the group acts on a vector space, such as the three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$. A representation of a group ${\displaystyle G}$ on an ${\displaystyle n}$-dimensional real vector space is simply a group homomorphism ${\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )}$ from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.

A group action gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.

Galois groups

Main article: Galois group

Galois groups were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation ${\displaystyle ax^{2}+bx+c=0}$ are given by

$${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$$

Each solution can be obtained by replacing the ${\displaystyle \pm }$ sign by ${\displaystyle +}$ or ${\displaystyle -}$; analogous formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomials and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and roots similar to the formula above.

Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.

Finite groups

Main article: Finite group

A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. An important class is the symmetric groups ${\displaystyle \mathrm {S} _{N}}$, the groups of permutations of ${\displaystyle N}$ objects. For example, the symmetric group on 3 letters ${\displaystyle \mathrm {S} _{3}}$ is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group ${\displaystyle \mathrm {S} _{N}}$ for a suitable integer ${\displaystyle N}$, according to Cayley's theorem. Parallel to the group of symmetries of the square above, ${\displaystyle \mathrm {S} _{3}}$ can also be interpreted as the group of symmetries of an equilateral triangle.

The order of an element ${\displaystyle a}$ in a group ${\displaystyle G}$ is the least positive integer ${\displaystyle n}$ such that ${\displaystyle a^{n}=e}$, where ${\displaystyle a^{n}}$ represents

$${\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}$$

that is, application of the operation "${\displaystyle \cdot }$" to ${\displaystyle n}$ copies of ${\displaystyle a}$. (If "${\displaystyle \cdot }$" represents multiplication, then ${\displaystyle a^{n}}$ corresponds to the ${\displaystyle n}$th power of ${\displaystyle a}$.) In infinite groups, such an ${\displaystyle n}$ may not exist, in which case the order of ${\displaystyle a}$ is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.

More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group ${\displaystyle G}$ the order of any finite subgroup ${\displaystyle H}$ divides the order of ${\displaystyle G}$. The Sylow theorems give a partial converse.

The dihedral group ${\displaystyle \mathrm {D} _{4}}$ of symmetries of a square is a finite group of order 8. In this group, the order of ${\displaystyle r_{1}}$ is 4, as is the order of the subgroup ${\displaystyle R}$ that this element generates. The order of the reflection elements ${\displaystyle f_{\mathrm {v} }}$ etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups ${\displaystyle \mathbb {F} _{p}^{\times }}$ of multiplication modulo a prime ${\displaystyle p}$ have order ${\displaystyle p-1}$.

Finite abelian groups

Any finite abelian group is isomorphic to a product of finite cyclic groups; this statement is part of the fundamental theorem of finitely generated abelian groups.

Any group of prime order ${\displaystyle p}$ is isomorphic to the cyclic group ${\displaystyle \mathrm {Z} _{p}}$ (a consequence of Lagrange's theorem). Any group of order ${\displaystyle p^{2}}$ is abelian, isomorphic to ${\displaystyle \mathrm {Z} _{p^{2}}}$ or ${\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}$. But there exist nonabelian groups of order ${\displaystyle p^{3}}$; the dihedral group ${\displaystyle \mathrm {D} _{4}}$ of order ${\displaystyle 2^{3}}$ above is an example.

Simple groups

When a group ${\displaystyle G}$ has a normal subgroup ${\displaystyle N}$ other than ${\displaystyle \{1\}}$ and ${\displaystyle G}$ itself, questions about ${\displaystyle G}$ can sometimes be reduced to questions about ${\displaystyle N}$ and ${\displaystyle G/N}$. A nontrivial group is called simple if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the Jordan–Hölder theorem.

Classification of finite simple groups

Main article: Classification of finite simple groups

Computer algebra systems have been used to list all groups of order up to 2000. But classifying all finite groups is a problem considered too hard to be solved.

The classification of all finite simple groups was a major achievement in contemporary group theory. There are several infinite families of such groups, as well as 26 "sporadic groups" that do not belong to any of the families. The largest sporadic group is called the monster group. The monstrous moonshine conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions.

The gap between the classification of simple groups and the classification of all groups lies in the extension problem.

Groups with additional structure

An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set ${\displaystyle G}$ equipped with a binary operation ${\displaystyle G\times G\rightarrow G}$ (the group operation), a unary operation ${\displaystyle G\rightarrow G}$ (which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs.

This way of defining groups lends itself to generalizations such as the notion of group object in a category. Briefly, this is an object with morphisms that mimic the group axioms.

Topological groups

The unit circle in the complex plane under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom).

 

Main article: Topological group

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, ${\displaystyle g\cdot h}$ and ${\displaystyle g^{-1}}$ must not vary wildly if ${\displaystyle g}$ and ${\displaystyle h}$ vary only a little. Such groups are called topological groups, and they are the group objects in the category of topological spaces. The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other topological field, such as the field of complex numbers or the field of p-adic numbers. These examples are locally compact, so they have Haar measures and can be studied via harmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in infinite Galois theory. A generalization used in algebraic geometry is the étale fundamental group.

Lie groups

Main article: Lie group

A Lie group is a group that also has the structure of a differentiable manifold; informally, this means that it looks locally like a Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth.

A standard example is the general linear group introduced above: it is an open subset of the space of all ${\displaystyle n}$-by-${\displaystyle n}$ matrices, because it is given by the inequality

$${\displaystyle \det(A)\neq 0,}$$

where ${\displaystyle A}$ denotes an ${\displaystyle n}$-by-${\displaystyle n}$ matrix.

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time, are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example is the group of Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of spacetime in special relativity. The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. An important example of a gauge theory is the Standard Model, which describes three of the four known fundamental forces and classifies all known elementary particles.

Generalizations

Group-like structures
	Totality	Associativity	Identity	Division	Commutativity
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Unital magma	Required	Unneeded	Required	Unneeded	Unneeded
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Loop	Required	Unneeded	Required	Required	Unneeded
Monoid	Required	Required	Required	Unneeded	Unneeded
Group	Required	Required	Required	Required	Unneeded
Commutative monoid	Required	Required	Required	Unneeded	Required
Abelian group	Required	Required	Required	Required	Required
^α The closure axiom, used by many sources and defined differently, is equivalent.

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers ${\displaystyle \mathbb {N} }$ (including zero) under addition form a monoid, as do the nonzero integers under multiplication ${\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}$, see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as ${\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )}$ is derived from ${\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}$, known as the Grothendieck group. Groupoids are similar to groups except that the composition ${\displaystyle a\cdot b}$ need not be defined for all ${\displaystyle a}$ and ${\displaystyle b}$. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e., an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group. The table gives a list of several structures generalizing groups.