For $H$ a group and $n\in\mathbb{N}$, let $H^{(n)}=\langle h^n : h \in H \rangle$. Now let $G$ be an extraspecial $p$-group (see definition). Is it true that $G^{(p)}\cong \mathbb{Z_p}$. (It holds for $D_8$ and $Q_8$.)
$p$-th powers of elements of an extraspecial $p$-group
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group-theory
finite-groups