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Perfectness is not subgroup-closed


It is possible to have a perfect group G and a subgroup H of G that is not perfect.

In fact, the following somewhat stronger statement is true: for any nontrivial perfect group G, we can find a subgroup H that is not perfect. Note that since nontrivial perfect groups do exist (for instance, alternating group:A5) this statement is indeed stronger.

Related facts


Given: A nontrivial perfect group G

To prove: G has a subgroup H that is not perfect.

Proof: Take any non-identity element of G, and define H as the cyclic subgroup generated by that element. H is a nontrivial cyclic group, and since cyclic implies abelian, it is a nontrivial abelian group, hence not perfect.