Perfectness is not subgroup-closed

This article gives the statement, and possibly proof, of a group property (i.e., perfect group) not satisfying a group metaproperty (i.e., subgroup-closed group property).
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Statement

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

• Perfectness is not characteristic subgroup-closed
• Every finite group is a subgroup of a finite perfect group
• Every group is a subgroup of a perfect group

Proof

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.