Permutation - Permutations in Group Theory

Permutations in Group Theory

In group theory, the term permutation of a set means a bijective map, or bijection, from that set onto itself. The set of all permutations of any given set S forms a group, with composition of maps as product and the identity as neutral element. This is the symmetric group of S. Up to isomorphism, this symmetric group only depends on the cardinality of the set, so the nature of elements of S is irrelevant for the structure of the group. Symmetric groups have been studied most in the case of a finite sets, in which case one can assume without loss of generality that S={1,2,...,n} for some natural number n, which defines the symmetric group of degree n, written S_n.

Any subgroup of a symmetric group is called a permutation group. In fact by Cayley's theorem any group is isomorphic to some permutation group, and every finite group to a subgroup of some finite symmetric group. However, permutation groups have more structure than abstract groups, allowing for instance to define the cycle type of an element of a permutation group; different realizations of a group as a permutation group need not be equivalent for this additional structure. For instance S₃ is naturally a permutation group, in which any transposition has cycle type (2,1), but the proof of Cayley's theorem realizes S₃ as a subgroup of S₆ (namely the permutations of the 6 elements of S₃ itself), in which permutation group transpositions get cycle type (2,2,2). So in spite of Cayley's theorem, the study of permutation groups differs from the study of abstract groups.