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

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 $|Z(G)| \le p^{2a}$.