Construction of the Outer Automorphism of {mathcal {S}}_{6} via a Complex Hadamard Matrix

Neil I Gillespie,Padraig Ó Catháin,Cheryl E Praeger

doi:10.1007/s11786-018-0382-0

Abstract

We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers.

Highlights

IntroductionSylvester showed that the fifteen two-subsets of a six element set can be formed into 5 parallel classes in six different ways and that the action of S6 on these synthematic totals is essentially different from its natural action on six points [13]
Sylvester showed that the fifteen two-subsets of a six element set can be formed into 5 parallel classes in six different ways and that the action of S6 on these synthematic totals is essentially different from its natural action on six points [13]. To our knowledge this was the first construction for the outer automorphism of S6
Miller attributes the result that for n = 6, Sn has no outer automorphisms to Hölder, and Sylvester’s construction of the outer automorphism of S6 to Burnside [11]

Summary

Introduction

Sylvester showed that the fifteen two-subsets of a six element set can be formed into 5 parallel classes in six different ways and that the action of S6 on these synthematic totals is essentially different from its natural action on six points [13]. To our knowledge this was the first construction for the outer automorphism of S6. Cameron and van Lint devoted an entire chapter (their sixth!) to the outer automorphism of S6 [2] They build on Sylvester’s construction to construct the 5-(12, 6, 1) Witt design, the projective plane of order 4, and the Hoffman–Singleton graph. It was in the latter paper that we first became aware of the complex Hadamard matrix of order 6 discussed in this article, where it is described as corresponding to the distance transitive triple cover of the complete bipartite graph K6,6

Hadamard Matrices

The Outer Automorphism of S6