Symmetric groups on finite sets are complete

Statement
For $$n \ne 2,6$$, the fact about::symmetric group $$\operatorname{Sym}(n)$$ on a set of size $$n$$ (i.e., the fact about::symmetric group on finite set), is a fact about::complete group: it is centerless and every automorphism of it is inner.

For $$n = 2$$, the group is not centerless, but every automorphism is inner.

For $$n = 6$$, the group is centerless, but not every automorphism is inner. In fact, the symmetric group of degree six is of index two in its automorphism group.

Related facts

 * Symmetric groups on infinite sets are complete
 * Automorphism group of alternating group equals symmetric group: This again holds under the same assumptions: $$n \ne 2,6$$.

Facts used

 * 1) Symmetric groups are centerless (for the centerlessness part)
 * 2) uses::Conjugacy class of transpositions is preserved by automorphisms
 * 3) uses::Transposition-preserving automorphism of symmetric group is inner

Proof

 * 1) Centerless: The fact that the symmetric group is centerless for $$n \ne 2$$ follows from fact (1).
 * 2) Every automorphism is inner: Fact (2) yields that every automorphism preserves the conjugacy class of transpositions when $$n \ne 6$$ follows from fact (2). Fact (3) then yields that, in fact, every automorphism is inner.