A permutation group is a finite group 
whose elements are permutations
of a given set and whose group operation is composition of permutations
in 
.
Permutation groups have orders dividing 
.
Two permutations form a group only if one is the identity element and the other is a permutation involution, i.e., a permutation which is its own inverse (Skiena 1990, p. 20). Every permutation group with more than two elements can be written as a product of transpositions.
Permutation groups are represented in the Wolfram Language as a set of permutation cycles with PermutationGroup. A set of permutations may be tested to see if it forms a permutation group using PermutationGroupQ[l] in the Wolfram Language package Combinatorica` .
Conjugacy classes of elements which are interchanged in a permutation group are called permutation cycles.
Examples of permutation groups include the symmetric group 
(of order 
),
the alternating group 
(of order 
for 
), the cyclic group 
(of order 
), and the dihedral group 
(of order 
).
