In mathematics, a finite group is a group which has finitely many elements. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups. It is too much to hope for a complete theory: the complexity becomes overwhelming when the group is large.
Less overwhelming, but still of interest, are some of the smaller general linear groups over finite fields. The group theorist J. L. Alperin has written that "The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled." (Bulletin (New Series) of the American Mathematical Society, 10 (1984) 121) For a discussion of one of the smallest such groups, GL(2,3), see Visualizing GL(2,p).
Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations. It turns out that continuous symmetry, as modelled by Lie groups, also leads to finite groups, the Weyl groups. In this way, finite groups and their properties can enter centrally in questions, for example in theoretical physics, where their role is not initially obvious.
- Lagrange's theorem
- Cauchy's theorem (group theory)
- Sylow theorems
- List of small groups
- Character theory
- Representation theory of finite groups
- Modular representation theory
- Classification of finite simple groups
- Monstrous moonshine
- Pro-finite group
- infinite group theory