An: Aut(An) ≅ Sn. Conjugation by even elements are inner automorphisms of An while the outer automorphism of An of order 2 corresponds to conjugation by an odd element. For , there is an exceptional outer automorphism of An so Sn is not the full automorphism group of An. Conversely, for , Sn has no outer automorphisms, and for it has no center, so for it is a complete group, as discussed in automorphism group, below. For , Sn is an almost simple group, as it lies between the simple group An and its group of automorphisms. Sn can be embedded into An+2 by appending the transposition to all odd permutations, while embedding into An+1 is impossible for . Generators and relations The symmetric group on letters is generated by the adjacent transpositions that swap and . The collection generates subject to the following relations: for , and where 1 represents the identity permutation. This representation endows the symmetric group with the structure of a Coxeter group (and so also a reflection group). Other possible generating sets include the set of transpositions that swap and for , and a set containing any -cycle and a -cycle of adjacent elements in the -cycle. Subgroup structure A subgroup of a symmetric group is called a permutation group. Normal subgroups The normal subgroups of the finite symmetric groups are well understood. If , Sn has at most 2 elements, and so has no nontrivial proper subgroups. The alternating group of degree n is always a normal subgroup, a proper one for and nontrivial for ; for it is in fact the only nontrivial proper normal subgroup of Sn, except when where there is one additional such normal subgroup, which is isomorphic to the Klein four group. The symmetric group on an infinite set does not have a subgroup of index 2, as Vitali (1915) proved that each permutation can be written as a product of three squares. However it contains the normal subgroup S of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of S that are products of an even number of transpositions form a subgroup of index 2 in S, called the alternating subgroup A. Since A is even a characteristic subgroup of S, it is also a normal subgroup of the full symmetric group of the infinite set. The groups A and S