Permutation group

It is easily shown that the set of permutations on a set of objects form a group, called a permutation group, with composition as the group operation. For example, let $$A$$ denote a finite set of $$n$$ distinct objects, and let $$S_A$$ denote the set of permutations of the elements of $$A$$. The criteria of associativity and the existence of inverses are obvious from the definition of permutations as bijections from $$A$$ to itself. The existence of an identity is slightly more difficult to establish, but we can define an identity mapping $$i(a)=a,\forall a\in A$$, and it is clear that this mapping is both a permutation and an identity.

Since all permutation groups over sets with the same number of elements are isomorphic, and since abstract algebra is generally only concerned with groups, rings, fields et cetera up to isomorphism, the generic permutation group over $$n$$ elements is simply denoted $$S_n$$.

A symmetry group is in general a subgroup of the permutation group over the number of vertices in the figure in question, since all affine transformations of a figure are permutations of its vertices, although the converse does not in general hold.