Automorphisms of The Symmetric and Alternating Groups - The Exceptional Outer Automorphism of S₆ - Other Constructions

Other Constructions

Ernst Witt found a copy of Aut(S₆) in the Mathieu group M₁₂ (a subgroup T isomorphic to S₆ and an element σ that normalizes T and acts by outer automorphism). Similarly to S₆ acting on a set of 6 elements in 2 different ways (having an outer automorphism), M₁₂ acts on a set of 12 elements in 2 different ways (has an outer automorphism), though since M₁₂ is itself exceptional, one does not consider this outer automorphism to be exceptional itself.

The full automorphism group of A₆ appears naturally as a maximal subgroup of the Mathieu group M₁₂ in 2 ways, as either a subgroup fixing a division of the 12 points into a pair of 6-element sets, or as a subgroup fixing a subset of 2 points.

Another way to see that S₆ has a nontrivial outer automorphism is to use the fact that A₆ is isomorphic to PSL₂(9), whose automorphism group is the projective semilinear group PΓL₂(9), in which PSL₂(9) is index 4, yielding an outer automorphism group of order 4. The most visual way to see this automorphism is to give an interpretation via algebraic geometry over finite fields, as follows. Consider the action of S₆ on affine 6-space over the field k with 3 elements. This action preserves several things: the hyperplane H on which the coordinates sum to 0, the line L in H where all coordinates coincide, and the quadratic form q given by the sum of the squares of all 6 coordinates. The restriction of q to H has defect line L, so there is an induced quadratic form Q on the 4-dimensional H/L that one checks is non-degenerate and non-split. The zero scheme of Q in H/L defines a smooth quadric surface X in the associated projective 3-space over k. Over an algebraic closure of k, X is a product of two projective lines, so by a descent argument X is the Weil restriction to k of the projective line over a quadratic etale algebra K. Since Q is not split over k, an auxiliary argument with special orthogonal groups over k forces K to be a field (rather than a product of two copies of k). The natural S₆-action on everything in sight defines a map from S₆ to the k-automorphism group of X, which is the semi-direct product G of PGL₂(K) = PGL₂(9) against the Galois involution. This map carries the simple group A₆ nontrivially into (hence onto) the subgroup PSL₂(9) of index 4 in the semi-direct product G, so S₆ is thereby identified as an index-2 subgroup of G (namely, the unique such subgroup distinct from PGL₂(9) that also does not contain the Galois involution). Conjugation by any element of G outside of S₆defines the nontrivial outer automorphism of S₆.