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Please help me about the following question: Suppose $G$ is a finite 2-group and $x\in Z(M)\setminus Z(G)$ for some maximal subgroup $M$ of $G$ such that $x^2\in Z(G)$, is it true $x\in Z_2(G)$? Whe …

Thanks for any comment or answer. Let $G$ be a finite non-abelian $p$-group such that $G=AB$ where $A=C_G(a)$ and $B=C_G(b)$ are maximal abelian normal subgroups of $G$ such that $A\cap B=Z(G)$, an …

I am interested in the nonabelian finite $p$-group $G$ with the following property: $G$ has a maximal abelian subgroup $A$ and there exists an $x\in G\setminus A$ such that $x$ normalize $A$ but $x$ …

Thanks for any help or comments. Suppose $G$ is a meta cyclic p-group, i.e. $G$ is an extension of cyclic by cyclic group, Is it true that every nonabelian maximal subgroup of $G$ is meta cyclic?

Thanks for any comment or answer. Suppose $G$ is a finite group. Then we call $H$ is a $p$-local subgroup of $G$ if $H=N_G(P)$ for some $p$-subgroup $P$ of $G$. My question is: Is it possible to cha …