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
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 Erd÷s-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
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
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
Groups acting on other algebraic objects are the symmetry groups of graphs,
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).
- 20A: Foundations
- 20B: Permutation
- 20C: Representation
theory of groups. See also 19A22 for representation rings and Burnside rings
Abstract finite groups
- 20E: Structure
and classification of infinite or finite groups
- 20F: Special
aspects of infinite or finite groups
- 20G: Linear
algebraic groups (classical groups). For arithmetic theory, see 11E57, 11H06;
for geometric theory, See 14LXX, 22EXX; for other methods in representation
theory, see 15AQ30, 22EXX, 15A30, 22E45, 22E46, 22E47, 22E50, 22E55
- 20H: Other
groups of matrices See also 15A30.
- 20J: Connections
with homological algebra and category theory
- 20K: Abelian
- 20L05: Groupoids
(i.e. small categories in which all morphisms are isomorphisms) For sets
with a single binary operation, see 20N02; for topological groupoids, see
- 20M: Semigroups
- 20N: Other
generalizations of groups
- 20P05: Probabilistic
methods in group theory
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
(old) classifications for this area at the AMS.
There are many textbooks in group theory.
- At the undergraduate level, group theory is typically presented
with other branches of abstract algebra, although group theory often dominates.
Classic examples include I. N. Herstein's books, the last being "Abstract
Algebra" (3rd edition 1996, Prentice Hall, Inc., Upper Saddle River, NJ,
249pp, ISBN 0-13-374562-7).
- A possible text specific to group theory might be that
of Marshall Hall, Jr. ("The theory of groups", Chelsea Publishing Co., New
York, 1976. 434 pp.)
More substantial texts at the graduate level include a number
restricted to finite
group theory (or some other subfield such as representation theory), and,
- Rotman, Joseph J., "An introduction to the theory of groups",
Fourth edition. Springer-Verlag, New York, 1995. 513 pp. ISBN 0-387-94285-8,
- Robinson, Derek J. S. "A course in the theory of groups",
Second edition. Springer-Verlag, New York, 1996. 499 pp. ISBN 0-387-94461-3
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:
- Passman, D. S.: "What is a group ring?", Amer. Math. Monthly
83 (1976), no. 3, 173--185. MR52#10859
- Farb, Benson: "Automatic groups: a guided tour", Enseign.
Math. (2) 38 (1992), no. 3-4, 291--313. MR93k:20052
- There is a Featured Review by Hyman Bass (MR95m:20054)
of a paper of Alexander Lubotzky ("Subgroup growth and congruence subgroups",
Invent. Math. 119 (1995), no. 2, 267--295) which surveys recent themes in
Unique resources include
- "Reviews on infinite groups", classified by Gilbert Baumslag.
American Mathematical Society, Providence, 1974. 1062 pages in two parts.
Reviews reprinted from Mathematical Reviews, Vols. 1--40 (Published during
1940--1970). For reviews of articles specifically on finite groups see Finite Group
Theory for a companion volume.
- Weinstein, Michael: "Examples of groups", Polygonal Publishing
House, Passaic, N.J., 1977. 307 pp.
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, email@example.com,
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
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:
- GAP (Groups, Algorithms
and Programming) is a general purpose group theory program.
(and its predecessor Cayley), a programming language designed for the investigation
of algebraic and combinatorial structures.
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
a control program for a collection of programs for working with automatic
- There has been some success proving results using the automated-reasoning
program Otter. (See
e.g. MR 94g:03025)
You can reach this page through http://www.math-atlas.org/welcome.html
- The crystallographic
- Under what conditions will an automorphism
of a subgroup extend to a larger group?
- Automorphisms and semidirect products
- Under what conditions will a representation
of a subgroup extend to a larger group? (Applications to Rubik's cube).
- Does every group have a two-, or even finite-dimensional
- Direct correspondence between irreducible representations
and conjugacy classes in finite groups
characters of symplectic groups in characteristic p
- A Brauer pair
(groups indistinguishable by character theory)
- What is the representation of a group induced by a
representation of a subgroup?
- Components of a representation
induced from a non-normal subgroup
- What are the finite subgroups
of the group of rotations in R^n?
- Stabilizers of
normal series are nilpotent.
- What are nilpotent groups
(and why that name?)
- The Burnside problem:
if every element of G has a bounded order, and G is finitely-generated,
is G finite? (No)
- Status of the Burnside Problem:
if all elements have bounded order, when is the group finite?
graph of symmetry group of icosahedron is a soccer ball (football)
- How many groups satisfy a particular set of relations
among their generators?
- Infinite torsion groups (all elements have finite
- Under what conditions is there in G a subgroup isomorphic
to the quotient group G/N ?
groups -- those with a subgroup H not meeting any of its conjugates.
- Finding generators for the
modular group PSL(2,Z).
- Is there an infinite group with only finitely many
conjugacy classes? (yes)
- The Word
Problem: can we decide if a specific presentation is of the trivial
group? (no) [Derek Holt]
- Finitely presented groups with solvable word
- Citations and summary of what's possible with computational
algorithm, replacing generators of a group with "strong generators"
so as to be able to determine group order, etc.
generating sets and the Schreier-Sims algorithm
- Is subgroup membership
(e.g.) effectively computable? (yes)
- Connections between representations
of Sym(n) and GL(n).
- Representations of crystallographic
- Sample problem in representations of finite groups solved
with techniques of semi-simple algebras.
- Structure theory for arbitrary Abelian groups
homomorphisms into G/Torsion to homomorphisms into G (G abelian)
- Constructing split extensions
with certain properties.
- What is the cohomology of groups
and how is it used to enumerate group extensions?
- What is a stem
cover of a group G ?
- Do there exist groups with many zero cohomology
- What is the projective dimension
of a Z[G]-module?
- Nilpotent elements in group algebras
for symmetric groups, representation theory
- What are the seven 1-dimensional symmetry groups?
- [Offsite] A good web page devoted to the 17 wallpaper
- Normal subgroups of infinite symmetric
groups including the corresponding alternating groups.
- Fairest algorithm to generate random permutation
- Can one permute the entries in an n x m grid by just permuting the
rows and columns? (No.)
- Use of permutation groups to determine a method for transposing
nonsquare matrices in place.
- Does a^3b^3=(ab)^3 mean G is abelian?
- Must a p-group have a non-trivial
center? (not if infinite)
- [Offsite] A page of some links of Group
theory for physicists
- Is there an algorithm to determine whether or not a group
is trivial (or more generally, whether two are isomorphic)? -- No.
method for computations in finitely-presented groups.
groups and free Abelian groups.
- Nielsen's theorem: subgroups of free
groups are free.
- Computing and applying free products
- Subgroups of
free groups always have a conjugate with which they have trivial intersection
- How to glue together
dodecahedra face-to-face to make a closed loop? (freeness of an extension
of the Dodecahedral group)
groups resemble linear groups and finite groups
- What are Borel subgroups
of algebraic groups?
- New preprint server
for Linear Algebraic Groups and Related Structures
- Simplicity of linear groups
- Determining symmetry groups
of a family of functions
Refinement Theorem for modular lattices (and thus to groups etc.)
- Equivalences among some varieties of
groups generated by a single group
- Which groups embed in their Bohr compactification?
- An ordered
group with least upper property is the integers or reals.
- What are the possible semigroup structures
on the real line?
between the additive and multiplicative semigroup structures on the real
line (e.g. the logarithm).
- Number of
semigroups of a given order?
- Endomorphisms of a semigroup
of linear transformations
- What is a loop? (sort of
a non-associative group)
- Smallest non-associative pseudogroup
- Some definitions and questions involving quasigroups, loops,