On automorphisms of products of groups

Summary

Abstract

When the finite group G can be written as a product G = PQ (a direct, semidirect, or central product), with Q ⊲ G, we investigate the extent to which Aut(P) and Aut(Q) figure in the structure of Aut(G). In particular, we study the image of the map p : Aut(G;Q) → Aut(G/Q) × Aut(Q), where Aut(G;Q) is the subgroup of automorphisms of G that restrict to automorphisms of Q.

Introduction

In this study, we are interested in the relationships among Aut(G) and the automorphisms of subgroups and subquotients of G. In particular, if G = PQ we would like to understand the extent to which Aut(P) and Aut(Q) figure in the structure of Aut(G). There is, a priori, no reason to believe that Aut(P) and Aut(Q) have any influence on Aut(G), but, indeed, there are conditions under which the influence can be both felt and described.

In the case that G = P × Q, the relationship among Aut(P), Aut(Q), and Aut(G) is easy to discern when one of P or Q is characteristic. More generally, there are conditions under which information on Aut(G/ Φ (G)) can be obtained from understanding the automorphisms of P, Q, or their Frattini quotients. For some algebraic topologists, this latter kind of information suffices for gaining knowledge about the stable homotopy decomposition of the classifying spaces of direct products of groups (see [3] and [6]).