On a Theorem of Hölder

Authors

• Donald W. Miller, University of Nebraska - Lincoln

Document Type

Article

Date of this Version

4-1958

Comments

Published in The American Mathematical Monthly, Vol. 65, No. 4 (Apr., 1958), pp. 252-254 Copyright 1958 Mathematical Association of America. Used by permission.

Abstract

A well-known result, due to Hölder [1], is the following: The symmetric group S_n, 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.