Let $G$ be a finite group such that $G'\cap Z(G)\neq 1$. Suppose also that $G'$ is an elementary abelian $p$-group; $G'\nleq Z(G) $; $(G/Z(G))'$ is a minimal normal subgroup of $G/Z(G)$.
Can we deduce that $(G/Z(G))'\cap Z(G/Z(G))\neq 1$?
2026-03-27 18:34:37.1774636477
A question about intersection of center and commutator subgroup
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No, we can't. Minimal counterexample: $G=\text{SmallGroup}(96,197)$.
In here, $G'\cong \mathbb{Z}_2\times\mathbb{Z}_2\times \mathbb{Z}_2$ and $Z(G)\cong \mathbb{Z}_2$.
$G/Z(G)\cong\text{SmallGroup}(48,49)$, and $(G/Z(G))'\cong \mathbb{Z}_2\times\mathbb{Z}_2$ is a minimal normal subgroup of $G/Z(G)$. We have that $Z(G/Z(G))\cong \mathbb{Z}_2\times\mathbb{Z}_2$ as well, but the two intersect trivially.