Centralizers of Camina $p$-groups of nilpotence class $3$

TL;DR

This paper investigates the structure of Camina p-groups of nilpotence class 3, establishing bounds on the center and conditions for centralizers based on subgroup configurations.

Contribution

It proves new bounds on the center of such groups and characterizes when the commutator subgroup is contained in its centralizer based on subgroup counts.

Findings

If G' < C_G(G'), then |Z(G)| ≤ |G':G_3|^{1/2}.

If G/G_3 has 1 or 2 abelian subgroups of order |G:G'|, then G' < C_G(G').

If G/G_3 has p^a + 1 abelian subgroups, then either G' < C_G(G') or |Z(G)| ≤ p^{2a}.

Abstract

Let $G$ be a Camina $p$-group of nilpotence class $3$. We prove that if $G' < C_G (G')$, then $|Z(G)| \le |G':G_3|^{1/2}$. We also prove that if $G/G_3$ has only one or two abelian subgroups of order $|G:G'|$, then $G' < C_G (G')$. If $G/G_3$ has $p^a + 1$ abelian subgroups of order $|G:G'|$, then either $G' < C_G (G')$ or $|Z(G)| \le p^{2a}$.