p-groups having a unique proper non-trivial characteristic subgroup

Details p-groups having a unique proper non-trivial characteristic subgroup p-groups having a unique proper non-trivial characteristic subgroup

2010-07-23

arxiv.org/abs/1007.4084v1

Mathematics

Group Theory

Scientific paper

We consider the structure of finite $p$-groups $G$ having precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$. The structure of $G$ varies markedly depending on whether $G$ has exponent $p$ or $p^2$, 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 $p^2$ for various values of $r$. The automorphism group induced on the Frattini quotient is, in various cases, related to a maximal linear group in Aschbacher's classification scheme.

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