During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.
During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.
Permutation groups 
The symmetric group SN describes all permutations of N letters. There are N! such permutations which gives the order of the group. By Cayley's theorem, any finite group can be expressed as a subgroup of a symmetric group for a suitable integer N. The alternating group is the subgroup consisting of only the even permutations.
Cyclic groups 
A cyclic group ZN is a group all of whose elements are powers of a particular element a where aN=a0=e, the identity. A typical realization of this group is as the complex N-th roots of unity. Sending a to a primitive root of unity gives an isomorphism between the two. This can be done with any finite cyclic group.
Groups of Lie type 
Main theorems 
Lagrange's theorem 
Sylow theorems 
This provides a partial converse to Lagrange's theorem giving information about how many subgroups of a given order are contained in G.
Number of groups of a given order 
Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are. Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. If n is the square of a prime, then there are exactly two possible isomorphism types of group of order n, both of which are abelian. If n is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order n, and the number grows very rapidly as the power increases.
Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when q < p are primes with p-1 not divisible by q. For a necessary and sufficient condition, see cyclic number.
If n is squarefree, then any group of order n is solvable. A theorem of William Burnside, proved using group characters, states that every group of order n is solvable when n is divisible by fewer than three distinct primes. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order n is solvable when n is odd.
For every positive integer n, most groups of order n are solvable. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the classification of finite simple groups. For any positive integer n there are at most two simple groups of order n, and there are infinitely many positive integers n for which there are two non-isomorphic simple groups of order n.
Table of distinct groups of order n 
|Order n||# Groups||Abelian||Non-Abelian|
See also 
- Association scheme
- Classification of finite simple groups
- List of finite simple groups
- Lagrange's theorem
- Cauchy's theorem (group theory)
- Abelian group
- Non-abelian group
- List of small groups
- Representation theory of finite groups
- Modular representation theory
- Monstrous moonshine
- Profinite group
- Infinite group theory
- Finite ring
- John F. Humphreys, A Course in Group Theory, Oxford University Press, 1996, pp. 238-242.