THE CONVERSE OF A THEOREM OF GASCHUTZ,W. ON FRATTINI SUBGROUPS

The Frattini subgroup phi(G) of a group G is the intersection of all m aximal subgroups of G if there exists a maximal subgroup and it is G i tself otherwise. As part of a program to construct all finite groups, W. Gaschutz characterised the Frattini factor groups of finite groups and gave information about the structure of Frattini subgroups themsel ves in terms of their automorphism groups (see [4]). Although he only stated his result in the context of finite groups, the proof of Satz 1 1 in [4] yields the following more general conclusion. Here Aut(N) is the group of automorphisms of the group N and Inn(N) is the group of i nner automorphisms.