# Perfectness is not characteristic 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., characteristic subgroup-closed group property).

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## Statement

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

## Related facts

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

## Proof

- Let be special linear group:SL(2,5).
- Let be the center of special linear group:SL(2,5), i.e., the center of . Since center is characteristic, is characteristic in .
- is not an abelian group.