Finite Group

A finite group is a group having finite group order. Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, and so on.

Properties of finite groups are implemented in the Wolfram Language as FiniteGroupData[group, prop].

The classification theorem of finite groups states that the finite simple groups can be classified completely into one of five types.

A convenient way to visualize groups is using so-called cycle graphs, which show the cycle structure of a given abstract group. For example, cycle graphs of the 5 nonisomorphic groups of order 8 are illustrated above (Shanks 1993, p. 85).

Frucht's theorem states that every finite group is the graph automorphism group of a finite undirected graph.

The finite (cyclic) group forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University mathematics department a capella group "The Klein Four."

The following table gives the numbers and names of the distinct groups of group order for small . In the table, denotes an cyclic group of group order , a group direct product, a dihedral group, the quaternion group, an alternating group, the non-Abelian finite group of order 12 that is not and not (and is not the purely rotational subgroup of the point group ), the quasihedral (or semihedral) group of order 16 with group presentation , the modular group of order 16 with group presentation , the group of order 16 with group presentation , the group of order 16 with group presentation , the group with group presentation , the generalized quaternion group of order 16 with group presentation , a symmetric group, the semidirect product of with with group presentation , the Frobenius group of order , the semidirect product of by with group presentation , the group with group presentation , the group with group presentation , and the semidirect product of by with group presentation

	#	Abelian	#	non-Abelian	total
1	1		0	-	1
2	1		0	-	1
3	1		0	-	1
4	2	,	0	-	2
5	1		0	-	1
6	1		1		2
7	1		0	-	1
8	3	, ,	2	,	5
9	2	,	0	-	2
10	1		1		2
11	1		0	-	1
12	2	,	3	, ,	5
13	1		0	-	1
14	1		1		2
15	1		0	-	1
16	5	, , , ,	9	, , , , , , , ,	14
17	1		0	-	1
18	2	,	3	, ,	5
19	1		0	-	1
20	2	,