Centralizers of Camina p-groups of nilpotence class 3
AbstractLetGbe a Camina\hskip-0.853583pt{p}-group of nilpotence class 3. We prove that if{G^{\prime}\hskip-0.853583pt<\hskip-0.853583ptC_{G}(G^{\prime})}, then{|G_{3}|\leq|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)|\leq p^{2a}}.
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1907 ◽
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