Year: 2010
Type: preprint
Abstract: 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 $p^2$, 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.
Source: arXiv (Cornell University)
Authors: Glasby, S. P., Palfy, P. P., Schneider, Csaba
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2949895880