Please refer to the Glossary of group theory for the definitions of terms used throughout group theory.

See also list of group theory topics.

Table of contents |

2 Elementary introduction 3 Some useful theorems 4 Generalizations 5 Miscellany 6 External link |

It was Walter Van Dyck who in 1882 gave the modern definition of a group.

Other important mathematicians in this subject area includes Artin, Noether, Sylow, and many more.

In algebraic topology, groups are used to describe invariants of topological spaces (the name of the torsion subgroup of an infinite group shows the legacy of this field of endeavor). They are called "invariants" because they are defined in such a way that they don't change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups.

The concept of Lie group (named for mathematician Sophus Lie) is important in the study of differential equations and manifolds; they combine analysis and group theory and are therefore the proper objects for describing symmetries of analytical structures. Analysis on these and other groups is called harmonic analysis.

In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.

An understanding of group theory is also important in the physical sciences. In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. In physics, groups are important because they describe the symmetries which the law of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.

- Some basic results in elementary group theory
- Butterfly lemma
- Fundamental theorem on homomorphisms
- Jordan-Hölder theorem
- Krull-Schmidt theorem
- Lagrange's theorem
- Sylow theorems

- If we eliminate the requirement that every element have an inverse, then we get a monoid.
- If we additionally do not require an identity either, then we get a semigroup.
- Alternatively, if we relax the requirement that the operation be associative while still requiring the possibility of division, then we get a loop.
- If we additionally do not require an identity, then we get a quasigroup.
- If we don't require any axioms of the binary operation at all, then we get a magma.

Lie groups, algebraic groups and topological groups are examples of group objects: group-like structures sitting in a category other than the ordinary category of sets.

Abelian groups form the prototype for the concept of an abelian category, which has applications to vector spaces and beyond.

Formal group laws are certain formal power series which have properties much like a group operation.

James Newman summarized group theory as follows: