# Perfect core is homomorph-containing

From Groupprops

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., perfect core) always satisfies a particular subgroup property (i.e., homomorph-containing subgroup)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Contents

## Statement

Suppose is a group and is the perfect core of : it is the unique largest perfect subgroup of . Then, is a homomorph-containing subgroup of : given any homomorphism , we have .

## Related facts

## Facts used

- Perfectness is quotient-closed: The image of a perfect group under a surjective homomorphism is perfect.

## Proof

**Given**: A group with perfect core and a homomorphism .

**To prove**: .

**Proof**: is perfect by fact (1). By the definition of perfect core, any perfect subgroup of is contained in . Thus, .