On a Theorem of Hölder

Abstract

A well-known result, due to Hölder [1], is the following: The symmetric group Sn, has outer automorphisms if and only if n = 6. The classical proof of the existence of a class of outer automorphisms of S6, as formulated by Burnside [2], rests in part on the theory of primitive groups and entails extensive computation. In this note we offer a direct method for constructing such automorphisms. The author is grateful to Professor R. H. Bruck for raising this problem and for subsequent helpful remarks.