p-Groups with a unique proper non-trivial characteristic subgroup
Department or Administrative Unit
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.
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 of Algebra
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