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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 $x$ does not commute with any noncentral element of $A$.

$p$-groups with maximal class have this property since they have maximal abelian subgroup of order $p^2$. Any reference or comment or maybe partial characterization will be useful for me.

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    $\begingroup$ Assuming that the missing quantifier is "there exists", another example is the group of upper unitriangular matrices in ${\rm GL}(n,p)$, which does not have maximal class for $n>2$ (although the subgroup $\langle x,A \rangle$ has maximal class). $\endgroup$
    – Derek Holt
    Commented Apr 13, 2017 at 11:50
  • $\begingroup$ @Maryam The condition ``$x$ normalize $A$" is superfluous, since in nilpotent groups maximal abelian subgroups are normal. $\endgroup$
    – Alireza Abdollahi
    Commented Apr 14, 2017 at 3:27
  • $\begingroup$ @Alireza, I think the converse of your assertion is correct. I mean, every maximal normal abelian subgroups are maximal abelian. $\endgroup$
    – Maryam
    Commented Apr 14, 2017 at 6:44
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    $\begingroup$ @AlirezaAbdollahi That is not true, the dihedral group of order $16$ has a maximal abelian subgroup of order $4$ that is not normal. $\endgroup$
    – Derek Holt
    Commented Apr 14, 2017 at 9:24

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