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

Authors

Stephen P. Glasby, Central Washington University
P. P. Pálfy, Alfréd Rényi Institute of Mathematics
Csaba Schneider, Universidade de Lisboa

Document Type

Article

Department or Administrative Unit

Mathematics

Publication Date

12-15-2011

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 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.

Comments

This article was originally published in Journal of Algebra. The full-text article from the publisher can be found here.

Due to copyright restrictions, this article is not available for free download from ScholarWorks @ CWU.

Recommended Citation

Glasby, S. P., Pálfy, P. P., & Schneider, C. (2011). p-Groups with a unique proper non-trivial characteristic subgroup. Journal of Algebra, 348(1), 85–109. https://doi.org/10.1016/j.jalgebra.2011.10.005

Journal

Journal of Algebra

Copyright

Copyright © 2011 Elsevier Inc. All rights reserved.