Suppose that $$ 1 \to Z(G) \to G \to P \to 1 $$ is a short exact sequence of groups where $ P $ is perfect and $ Z(G) $ is the center of $ G $. Must it be the case that $ G $ is the direct product of a perfect group and an abelian group?
2026-03-26 13:01:36.1774530096
Group mod center is perfect structure result
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I think this has probably been answered in the comments, but the answer is no, and a counterexample is a central product of ${\rm SL}(2,5)$ with $C_4$, where $P={\rm PSL}(2,5) \cong A_5$, and $Z(G) \cong C_4$.