Let $G$ be a group , under what conditions do we have that $G/[G,G]$ is a finite group of prime order ?
2025-01-13 05:11:40.1736745100
When is the commutator subgroup a maximal subgroup?
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Hint: This happens if $G/[G,G]=Z/p$ where $p$ is a prime. The expression $G/[G,G]$ can also be seen has the first homology of $G$ in respect with $Z$. Thus, $[G,G]$ is maximal if and only if $H_1(G,Z)=Z/p$ where $p$ is a prime, which can be seen as a condition satisfied by $G$.