Centralizers of Camina p-groups of nilpotence class 3

Abstract

Abstract Let G be a Camina p \hskip-0.853583pt{p} -group of nilpotence class 3. We prove that if G ′ < C G ⁢ ( G ′ ) {G^{\prime}\hskip-0.853583pt<\hskip-0.853583ptC_{G}(G^{\prime})} , then | G 3 | ≤ | G ′ : G 3 | 1 / 2 {|G_{3}|\leq|G^{\prime}:G_{3}|^{1/2}} . We also prove that if G / G 3 {G/G_{3}} has only one or two abelian subgroups of order | G : G ′ | {|G:G^{\prime}|} , then G ′ < C G ⁢ ( G ′ ) {G^{\prime}<C_{G}(G^{\prime})} . If G / G 3 {G/G_{3}} has p a + 1 {p^{a}+1} abelian subgroups of order | G : G ′ | {|G:G^{\prime}|} , then either G ′ < C G ⁢ ( G ′ ) {G^{\prime}<C_{G}(G^{\prime})} or | Z ⁢ ( G ) | ≤ p 2 ⁢ a {|Z(G)|\leq p^{2a}} .