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

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