The exceptional symmetry

TL;DR

This paper provides an elementary proof that symmetric groups have a unique exceptional symmetry only in the case of S6, where the outer automorphism group is nontrivial, highlighting a special case in group theory.

Contribution

It offers a simple, elementary proof confirming the uniqueness of the exceptional symmetry of S6 among symmetric groups.

Findings

Outer automorphism group of S_n is trivial for all n except 6.

S6 has a unique nontrivial outer automorphism.

Elementary proof simplifies understanding of symmetric group symmetries.

Abstract

This note gives an elementary proof that the symmetric groups possess only one exceptional symmetry. I am referring to the fact that the outer automorphism group of the symmetric group $S_n$ is trivial unless $n=6$ and the outer automorphism group of $S_6$ has a unique nontrivial element.