A permutation group is a group 
whose elements are permutations
of a given set 
and whose group operation
is function composition. If 
is a finite set with 
elements, then 
is a subgroup of the symmetric
group 
,
so its group order divides 
by Lagrange's group
theorem.
Two distinct 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 element (Skiena 1990, p. 20). Every permutation of a finite set can be written as a product of transpositions, with the identity element represented by the empty product. These transpositions need not belong to the given permutation group.
Every permutation of a finite set has a decomposition into disjoint permutation
cycles. Two permutations belong to the same conjugacy class in the symmetric
group 
iff their permutation cycles
have the same lengths, counted with multiplicity.
Permutation groups are represented in the Wolfram Language with PermutationGroup, specified by a list of group generators given in disjoint permutation cycle notation.
Examples of finite permutation groups include the symmetric group 
(of group order 
), the alternating group 
(of group
order 
for 
),
the cyclic group 
(of group order 
), and the dihedral group 
(of group
order 
).
