# Perfectness is not subgroup-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., perfect group)notsatisfying a group metaproperty (i.e., subgroup-closed group property).

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## Statement

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

In fact, the following somewhat stronger statement is true: for *any* nontrivial perfect group , we can find a subgroup 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

**To prove**: has a subgroup that is not perfect.

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