The Outer Automorphism of S_6

Abstract

The symmetric groups are fundamental objects in abstract algebra, and for nearly every value of n, their automorphism groups are completely inner. The unique exception is the symmetric group S_6, which possesses a nontrivial outer automorphism. This paper develops the theory leading to this exception by introducing symmetric groups, automorphisms, and complete groups. After establishing why S_n is complete for all n̸ = 2, 6 and later proving it, we investigate the exceptional structure of S_6. We present several explicit constructions of its outer automorphism, including the six mystic pentagons, labeled triangles and labeled icosahedra as well as an intuitive interpretation arising from projective geometry. Each construction realizes the same automorphism by exhibiting a non-standard action of S_6 on a different six-element combinatorial or geometric object, showing that trans- positions are mapped to triple transpositions and therefore cannot arise from conjugation. This unexpected symmetry is one of the most remarkable phenomena in finite group theory.

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