The group of permutations of (that is the automorphism group of in ) is the symmetric group (or permutation group) on . This group may be denoted , , or . When is the finite set with elements, one typically writes or ; note that this group has elements.
In combinatorics, one often wants a slight generalisation. Given a natural number , an -permutation from is an injective function from to , that is a list of distinct elements of . Then an -permutation from is the same as a permutation of . (That an injective function from to itself must be invertible characterises as a Dedekind-finite set.)