Perfectness is not subgroup-closed

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.