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A group $G$ is perfect if it is equal to its own commutator subgroup $G'$. Equivalently, $G$ has no non-trivial cyclic (equivalently abelian, equivalently soluble) quotients. All non-abelian simple and quasisimple groups are perfect.

A perfect group may have cyclic composition factors, for example, $\mathrm{SL}(2,5)=C_2.A_5$ is perfect.

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• Last edited by Jennifer Paulhus on 2022-07-18 16:01:32
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