Endomorphism structure of symmetric groups

Automorphism structure

 * For $$n \ne 2,6$$, the symmetric group of degree $$n$$ is a complete group, i.e., the action on itself by conjugation gives an isomorphism from the group to its automorphism group. In particular, the symmetric group is isomorphic to its automorphism group.
 * For $$n = 2$$, the automorphism group is trivial.
 * For $$n = 6$$, i.e., for symmetric group:S6, the action on itself by conjugation induces an injective homomorphism to the automorphism group that is not surjective. The image has index two in the automorphism group. In other words, the outer automorphism group is cyclic group:Z2. See automorphism group of symmetric group:S6.

Other endomorphisms

 * For $$n = 0,1$$, every endomorphism is an automorphism.
 * For $$n = 2$$, every endomorphism is either trivial or an automorphism.
 * For $$n \ge 3$$ and $$n \ne 4$$, every endomorphism is of one of these three types: an automorphism, the trivial map, or a map with image a cyclic subgroup of order two, obtained by applying the sign homomorphism. For the last type of endomorphism, if the non-identity element in the image is an odd permutation, then the map is a retraction, whereas if it is an even permutation, it is not a retraction.
 * For $$n = 4$$, there are additional endomorphisms possible due to the presence of a normal V4 in S4.

Thus, for a generic $$n \ne 0,1,2,4,6$$, we have: