In
mathematics and
abstract algebra,
group theory studies the
algebraic structures known as
groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as
rings,
fields, and
vector spaces can all be seen as groups endowed with additional
operations and
axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.
Linear algebraic groups and
Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as
crystals and the
hydrogen atom, can be modelled by
symmetry groups. Thus group theory and the closely related
representation theory have many important applications in
physics,
chemistry, and
materials science. Group theory is also central to
public key cryptography.
One of the most important mathematical achievements of the 20th century
[1] was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete
classification of finite simple groups.
History
Group theory has three main historical sources:
number theory, the theory of
algebraic equations, and
geometry. The number-theoretic strand was begun by
Leonhard Euler, and developed by
Gauss's work on
modular arithmetic and additive and multiplicative groups related to
quadratic fields. Early results about
permutation groups were obtained by
Lagrange,
Ruffini, and
Abel in their quest for general solutions of polynomial equations of high degree.
Évariste Galois coined the term "group" and established a connection, now known as
Galois theory, between the nascent theory of groups and
field theory. In geometry, groups first became important in
projective geometry and, later,
non-Euclidean geometry.
Felix Klein's
Erlangen program proclaimed group theory to be the organizing principle of geometry.
Galois, in the 1830s, was the first to employ groups to determine the solvability of
polynomial equations.
Arthur Cayley and
Augustin Louis Cauchy pushed these investigations further by creating the theory of
permutation groups. The second historical source for groups stems from
geometrical situations. In an attempt to come to grips with possible geometries (such as
euclidean,
hyperbolic or
projective geometry) using group theory,
Felix Klein initiated the
Erlangen programme.
Sophus Lie, in 1884, started using groups (now called
Lie groups) attached to
analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in
algebraic number theory.
The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of
abstract algebra in the early 20th century,
representation theory, and many more influential spin-off domains. The
classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the
finite simple groups.
Main classes of groups
The range of groups being considered has gradually expanded from finite
permutation groups and special examples of
matrix groups to abstract groups that may be specified through a
presentation by generators and relations.
Permutation groups
The first class of groups to undergo a systematic study was
permutation groups. Given any set
X and a collection
G of
bijections of
X into itself (known as
permutations) that is closed under compositions and inverses,
G is a group
acting on
X. If
X consists of
n elements and
G consists of
all permutations,
G is the
symmetric group Sn; in general, any permutation group
G is a
subgroup of the symmetric group of
X. An early construction due to
Cayley exhibited any group as a permutation group, acting on itself (
X =
G) by means of the left
regular representation.
In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for
n ≥ 5, the
alternating group An is
simple, i.e. does not admit any proper
normal subgroups. This fact plays a key role in the
impossibility of solving a general algebraic equation of degree n ≥ 5 in radicals.
Matrix groups
The next important class of groups is given by
matrix groups, or
linear groups. Here
G is a set consisting of invertible
matrices of given order
n over a
field K that is closed under the products and inverses. Such a group acts on the
n-dimensional vector space
Kn by
linear transformations. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group
G.
Transformation groups
Permutation groups and matrix groups are special cases of
transformation groups: groups that act on a certain space
X preserving its inherent structure. In the case of permutation groups,
X is a set; for matrix groups,
X is a
vector space. The concept of a transformation group is closely related with the concept of a
symmetry group: transformation groups frequently consist of
all transformations that preserve a certain structure.
The theory of transformation groups forms a bridge connecting group theory with
differential geometry. A long line of research, originating with
Lie and
Klein, considers group actions on
manifolds by
homeomorphisms or
diffeomorphisms. The groups themselves may be
discrete or
continuous.
Abstract groups
Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a
presentation by
generators and relations,
A significant source of abstract groups is given by the construction of a
factor group, or
quotient group,
G/
H, of a group
G by a
normal subgroup H.
Class groups of
algebraic number fields were among the earliest examples of factor groups, of much interest in
number theory. If a group
G is a permutation group on a set
X, the factor group
G/
H is no longer acting on
X; but the idea of an abstract group permits one not to worry about this discrepancy.
The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under
isomorphism, as well as the classes of group with a given such property:
finite groups,
periodic groups,
simple groups,
solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of
abstract algebra in the works of
Hilbert,
Emil Artin,
Emmy Noether, and mathematicians of their school.
[citation needed]
Topological and algebraic groups
An important elaboration of the concept of a group occurs if
G is endowed with additional structure, notably, of a
topological space,
differentiable manifold, or
algebraic variety. If the group operations
m (multiplication) and
i (inversion),
are compatible with this structure, i.e. are
continuous,
smooth or
regular (in the sense of algebraic geometry) maps then
G becomes a
topological group, a
Lie group, or an
algebraic group.
[2]
The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for
abstract harmonic analysis, whereas
Lie groups (frequently realized as transformation groups) are the mainstays of
differential geometry and unitary
representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups.
Thus,
compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group
Γ can be realized as a
lattice in a topological group
G, the geometry and analysis pertaining to
G yield important results about
Γ. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (
profinite groups): for example, a single
p-adic analytic group G has a family of quotients which are finite
p-groups of various orders, and properties of
G translate into the properties of its finite quotients.
Combinatorial and geometric group theory
Groups can be described in different ways. Finite groups can be described by writing down the
group table consisting of all possible multiplications
g • h. A more compact way of defining a group is by
generators and relations, also called the
presentation of a group. Given any set
F of generators {
gi}
i ∈ I, the
free group generated by
F subjects onto the group
G. The kernel of this map is called subgroup of relations, generated by some subset
D. The presentation is usually denoted by
〈F | D 〉. For example, the group
Z = 〈a | 〉 can be generated by one element
a (equal to +1 or −1) and no relations, because
n·1 never equals 0 unless
n is zero. A string consisting of generator symbols and their inverses is called a
word.
Combinatorial group theory studies groups from the perspective of generators and relations.
[3] It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of
graphs via their
fundamental groups. For example, one can show that every subgroup of a free group is free.
There are several natural questions arising from giving a group by its presentation. The
word problem asks whether two words are effectively the same group element. By relating the problem to
Turing machines, one can show that there is in general no
algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the
group isomorphism problem, which asks whether two groups given by different presentations are actually isomorphic. For example the additive group
Z of integers can also be presented by
- 〈x, y | xyxyx = e〉;
it may not be obvious that these groups are isomorphic.
[4]
The Cayley graph of 〈 x, y ∣ 〉, the free group of rank 2.
Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.
[5] The first idea is made precise by means of the
Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the
word metric given by the length of the minimal path between the elements. A theorem of
Milnor and Svarc then says that given a group
G acting in a reasonable manner on a
metric space X, for example a
compact manifold, then
G is
quasi-isometric (i.e. looks similar from the far) to the space
X.
Representation of groups
Saying that a group
G acts on a set
X means that every element defines a bijective map on a set in a way compatible with the group structure. When
X has more structure, it is useful to restrict this notion further: a representation of
G on a
vector space V is a
group homomorphism:
- ρ : G → GL(V),
where
GL(
V) consists of the invertible
linear transformations of
V. In other words, to every group element
g is assigned an
automorphism ρ(
g) such that
ρ(g) ∘ ρ(h) = ρ(gh) for any
h in
G.
This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.
[6] On the one hand, it may yield new information about the group
G: often, the group operation in
G is abstractly given, but via
ρ, it corresponds to the
multiplication of matrices, which is very explicit.
[7] On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if
G is finite, it is
known that
V above decomposes into
irreducible parts. These parts in turn are much more easily manageable than the whole
V (via
Schur's lemma).
Given a group
G,
representation theory then asks what representations of
G exist. There are several settings, and the employed methods and obtained results are rather different in every case:
representation theory of finite groups and representations of
Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's
characters. For example,
Fourier polynomials can be interpreted as the characters of
U(1), the group of
complex numbers of
absolute value 1, acting on the
L2-space of periodic functions.
Connection of groups and symmetry
Given a structured object
X of any sort, a
symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example
- If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups.
- If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X.
- If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example.
- Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation
-
- has the two solutions
, and 
. In this case, the group that exchanges the two roots is the Galois group belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots.
The axioms of a group formalize the essential aspects of
symmetry. Symmetries form a group: they are
closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object.
Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative.
Frucht's theorem says that every group is the symmetry group of some
graph. So every abstract group is actually the symmetries of some explicit object.
The saying of "preserving the structure" of an object can be made precise by working in a
category. Maps preserving the structure are then the
morphisms, and the symmetry group is the
automorphism group of the object in question.
Applications of group theory
Applications of group theory abound. Almost all structures in
abstract algebra are special cases of groups.
Rings, for example, can be viewed as
abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore group theoretic arguments underlie large parts of the theory of those entities.
Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The
fundamental theorem of Galois theory provides a link between
algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding
Galois group. For example,
S5, the
symmetric group in 5 elements, is not solvable which implies that the general
quintic equation cannot be solved by radicals in the way equations of lower degree can.
The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as
class field theory.
Algebraic topology is another domain which prominently
associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of
topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some
deformation. For example, the
fundamental group "counts" how many paths in the space are essentially different. The
Poincaré conjecture, proved in 2002/2003 by
Grigori Perelman is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of
Eilenberg–MacLane spaces which are spaces with prescribed
homotopy groups. Similarly
algebraic K-theory stakes in a crucial way on
classifying spaces of groups. Finally, the name of the
torsion subgroup of an infinite group shows the legacy of topology in group theory.
A torus. Its abelian group structure is induced from the map
C → C/Z+τZ, where
τ is a parameter living in the
upper half plane.
Algebraic geometry and
cryptography likewise uses group theory in many ways.
Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures.
[8] The one-dimensional case, namely
elliptic curves is studied in particular detail. They are both theoretically and practically intriguing.
[9] Very large groups of prime order constructed in
Elliptic-Curve Cryptography serve for
public key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the
discrete logarithm very hard to calculate. One of the earliest encryption protocols,
Caesar's cipher, may also be interpreted as a (very easy) group operation. In another direction,
toric varieties are
algebraic varieties acted on by a
torus. Toroidal embeddings have recently led to advances in
algebraic geometry, in particular
resolution of singularities.
[10]
Algebraic number theory is a special case of group theory, thereby following the rules of the latter. For example,
Euler's product formula
captures
the fact that any integer decomposes in a unique way into
primes. The failure of this statement for
more general rings gives rise to
class groups and
regular primes, which feature in
Kummer's treatment of
Fermat's Last Theorem.
The circle of fifths may be endowed with a cyclic group structure
- In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group.
