20: Group Theory and Generalizations
Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way.
Formally, a group is a set G on which there is a multiplication '*' defined, satisfying the associative law. In addition, there is to be an element '1' in G with 1*g=g*1=g for every g in G; and every element g in G must have an inverse h satisfying g*h=h*g=1.
A particularly important class of groups is the set of permutation groups, those in which the elements are permutations of some set, and the group operation is simply composition. For example, the symmetric group on N objects is the set of all N! rearrangements of the N elements. Other important examples include the alternating groups and the Mathieu groups. In some sense, every group is a permutation group, but interesting questions arise in relation to the action on the set. For example, one considers groups which are highly transitive (they include enough symmetries to permute many large subsets), or groups which preserve additional structure of the set being permuted (angles in space, for example). Many combinatorial questions can be reduced to questions about the symmetric group; even the Rubik's cube can be viewed as a puzzle concerning a particular permutation group.
A second large class of groups are the linear groups (or "classical groups"). These are subgroups of the groups GL_n(R) of invertible n-by-n matrices over a field or ring R. Besides GL_n itself, this includes the group SL_n of matrices of determinant 1, the orthogonal group O_n, and related groups (such as the unitary groups) defined via the automorphisms of R. Other examples are provided by subrings of R, for example the integral matrices GL_n(Z), a discrete group. Crystallographers consider the groups of matrices which preserve a crystalline structure in space. These groups are even common in complex analysis, thanks to a related action they have on complex half-spaces. There is a well-developed literature on their internal structure (e.g. many of them are simple), representation theory (of course! since they are given as operating on a linear space), and so on. When the ring is the field of real or complex numbers, the group GL_n(R) has the additional structure of a manifold; groups with this extra structure are the Lie Groups studied in another section (22).
Another important class of groups is the set of Abelian groups, those whose elements commute. For example, this includes the additive groups of rings, fields, and vector spaces. Since the classification of finitely-generated Abelian groups is so precise, the interest is in discerning the structure of large abelian groups, and thus this area overlaps to a degree with set theory. Certain classes of abelian groups are more amenable to study -- those which are torsion, or torsion free, for example, or those which have some additional structure such as an ordering or a topology; this leads to the introduction of homological methods.
Of course it is also natural to study finite groups, not only because this appears to be a simpler topic but because they arise frequently in applications (e.g. as the symmetry groups of finite sets or polyhedra). Interestingly, finite group theory bears a resemblance to number theory as a consequence of Lagrange's theorem and the Sylow theorems. Thus one takes an interest in p-groups (groups of order a power of a prime) and the ways p-groups can interact in a group. As became clear during the twentieth century, this is a powerful avenue of investigation available for finite groups. A culminating feature of the analysis of finite groups is the proof of the classification of the "simple" ones (without normal subgroups), a result so deep that even an accurate succinct statement is difficult to achieve!
When studying groups, there are numerous internal features which can be examined -- particular subgroups and quotient groups, such as the center or abelianization; extensions which can be made with the group; the automorphism group or subgroup lattice or other similar constructs; maps between the one group and others. These lead to the clarification of types of groups of particular interest: we have already mentioned the Abelian and finite groups, but solvable groups, nilpotent groups, locally-finite groups, and a host of other classes arise naturally with respect to some internal properties under investigation.
Some groups have some additional structure. We have already mentioned those which have a given action on a set or a vector space. Another common setting is to have a group with a preferred set of generators; then one asks questions relating to the expression of elements of the group in terms of these generators (can we determine if a product of the generators is trivial? can we express an element in some minimal way?); this is sometimes known as "combinatorial group theory". It is closely related to the action of the group on combinatorial or geometric structures (the Cayley graph, say).
This leads to a number of connections to logic: one may ask for the existence of algorithms for groups based on their presentations in terms of generators and the relations among them. Can one tell whether a group is trivial (the word problem)? Do local conditions on the order determine global conditions (the Burnside problem)?
An important tool for the study of groups (particularly finite groups and compact groups) is representation theory. Broadly speaking, this asks for possible ways to view a group as a permutation group or a linear group. More narrowly, it considers homomorphisms from the group into the matrix groups GL_n(R) where R is most frequently the complex number field, or perhaps the ring of integers or a p-adic ring. Interest in this area is heightened by the fact that representations tend to be severely limited by the structure of the group, and that therefore any information about the representations provides information about the group. The information can be collected in a shorthand way ("character theory"). A number of attractive areas of representation theory link representations of a group with those of its subgroups, especially normal subgroups, algebraic subgroups, and local subgroups. Representation theory also considers images of groups in the automorphism groups of other abelian groups than simply complex vector spaces; these then are the group modules. (This is a somewhat more flexible setting than abstract group theory, since we move into an additive category); modular representation theory studies the case in which the modules are vector spaces over fields with positive characteristic.
Modern treatments of group theory include the use of tools from other categories such as rings and functors. This applies for example to representation theory, since we may attach homology and cohomology groups to modules for a group; this provides a method for distinguishing modules, constructing extensions, providing numerical invariants, and so on. It also focuses attention on projective modules. Taking trivial G-modules focuses attention on the group G itself, that is, group cohomology can be used to reflect the internal structure of G such as its p-rank. Since homology theory is rooted in topology, it can be used to study the possible ways a group can act on spaces or other sets with some structure.
"Probabilistic group theory" may be thought of the study of disordered systems: given a group, such as the group of symmetries of some structure, one can ask questions about randomly selected elements of that group: what is its expected order, how far is it from the identity (as a word in some string of generators) and so on. This leads to a number of Erdös-like problems, as well as applications to card-shuffling!
Algebraic systems with freer structure than groups include semigroups, monoids, and groupoids, all included in this section. They arise naturally in analysis and topology, among other areas. For example, the set of all curves in a topological space forms a groupoid under concatenation, that is, all the group axioms hold except a*b is not necessarily defined for all a and b. (Groupoids -- "small categories in which all morphisms are isomorphisms" -- are the only objects whose definition appears in the Mathematical Subject Classification!)
Semigroups, in particular, constitute a large family of mathematical objects. These are sets with associative binary operations but not necessarily inverses. (In some accounts, no identity is assumed either; those semigroups with an identity are "monoids".) In particular, the set of all maps from a set to itself (in some category) form a semigroup. In section 20, semigroups are studied from an algebraic perspective: one considers various classes of semigroups (free, regular, inverse), their structure theory (radicals, ideals); and their representations (as semigroups of endomorphisms, especially as linear semigroups).
Other generalizations of groups include loops (no associative law), quasigroups, hypergroups, and fuzzy groups.
The history written for the St Andrews archive is excellent for pre-20th-century developments.
Groups with additional structure include Topological Groups and Lie Groups. The study of analysis, and in particular differential equations, on these groups is Harmonic Analysis. Of course, a good deal of analysis (e.g. Fourier series) invokes the action of specific groups on the real line or complex plane.
There are also Ordered groups and Fuzzy groups.
The actions of groups on other mathematical objects give group theory links with many other branches of mathematics. In general, questions about these group actions tend to be treated as part of the other discipline. Groups acting on vector spaces are subgroups of the matrix groups studied in Linear Algebra. Groups acting on fields are Galois groups studied in Field Extensions. Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology. Groups acting on particular low-dimensional topological spaces give rise to aspects of Manifold Theory (e.g. knots). Groups acting on Euclidean space give us the structures of Geometry; in particular, there is a strong group-theoretic flavor to the study of the regular polyhedra. Groups acting on other algebraic objects are the symmetry groups of graphs, lattices, rings and modules, and so on. The actions of specific groups are used in applications such as Special Relativity.
We have mentioned that some axiomatic questions in group theory lead to mathematical logic and set theory.
For semigroups which arise in applications see Operator Theory and Global Analysis (for applications in analysis and cellular automata) or General Topology (for semigroups of transformations of general spaces).
We include with 20D some topics in permutation groups when the groups are obviously finite (e.g. the Rubik group), although these will eventually be moved to a separate section 20B.
Browse all (old) classifications for this area at the AMS.
There are many textbooks in group theory.
More substantial texts at the graduate level include a number restricted to finite group theory (or some other subfield such as representation theory), and, for example,
In general, textbook treatments of most of section 20 tend to cover the last few subfields only lightly or not at all.
Survey articles on various topics:
Unique resources include
A colleague has made available some worked-out exercises which can be used in conjunction with his text in abstract algebra. This covers elementary number theory, permutations, groups, rings, and fields.
There is a mailing list, firstname.lastname@example.org, for the discussion of any aspect of Group Theory. See also the Group Pub Forum Home Page, below.
There is a separate mailing list (with archive) for semigroup theory and related topics.
The standard tables for geometric group actions are the "International tables for crystallography"; Vol. A. is "Space-group symmetry", edited by Theo Hahn, D. Reidel Publishing Co., Dordrecht-Boston, 1983, 854 pp. ISBN 90-277-1445-2. Includes all 230 space groups.
An increasing number of group-theoretic features is incorporated into general-purpose symbolic programs. However, there are two programs specifically designed for heavy-duty investigations into group theory:
There are also programs suitable for investigations into particular limited areas of group theory. Those specific to questions for finite groups are on a separate page; we also have