Let $p$ be an odd prime. Does a nonabelian finite simple group $S$ exist such that $H^2(S, \mathbb{Z}/p\mathbb{Z})$ is not trivial?
2026-03-25 23:40:29.1774482029
A nonsplit extension of a nonabelian finite simple group by a cyclic group of odd prime order
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Yes, for example $S={\rm PSL}(p,q)$ where $p|q-1$. Then ${\rm SL}(p,q)$ is a non-split extension of $C_p$ by $S$.