p-Groups with a unique proper non-trivial characteristic subgroup

We consider the structure of finite p-groups G having precisely three characteristic subgroups, namely 1, Φ(G) and G. The structure of G varies markedly depending on whether G has exponent p or p2, and, in both cases, the study of such groups raises deep problems in representation theory. We present classification theorems for 3- and 4-generator groups, and we also study the existence of such r-generator groups with exponent p2 for various values of r. The automorphism group induced on the Frattini quotient is, in various cases, related to a maximal linear group in Aschbacherʼs classification scheme.