Finite group

Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics Glossary of group theory
Finite groups Classification of finite simple groups Cyclic group Z/nZ Symmetric group Sn Dihedral group Dn Alternating group An Mathieu groups M11 M12 M22 M23 M24 Conway groups Co1 Co2 Co3 Janko groups J1 J2 J3 J4 Fischer groups F22 F23 F24 Baby Monster group (B) Monster group (M)
Discrete groups Lattices Integers (Z) Lattice Modular groups PSL(2,Z) SL(2,Z)
Topological / Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Elliptic curve Linear algebraic group Abelian variety

In mathematics and abstract algebra, a finite group is a group whose underlying set G has a finite number of elements.

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.

Examples

Permutation groups

Main article: Permutation group

The symmetric group S_N 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

Main article: Cyclic group

A cyclic group Z_N is a group all of whose elements are powers of a particular element a where a^N=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 article: Group of Lie type

Main theorems

Lagrange's theorem

Main article: Lagrange's theorem (group theory)

For any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.

Sylow theorems

Main article: 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
1	1	1	0
2	1	1	0
3	1	1	0
4	2	2	0
5	1	1	0
6	2	1	1
7	1	1	0
8	5	3	2
9	2	2	0
10	2	1	1
11	1	1	0
12	5	2	3
13	1	1	0
14	2	1	1
15	1	1	0
16	14	5	9
17	1	1	0
18	5	2	3
19	1	1	0
20	5	2	3
21	2	1	1
22	2	1	1
23	1	1	0
24	15	3	12
25	2	2	0

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
• P-group
• List of small groups
• Representation theory of finite groups
• Modular representation theory
• Monstrous moonshine
• Profinite group
• Infinite group theory
• Finite ring

External references

• Number of groups of order n (sequence A000001 in OEIS)
• A classifier for groups of small order