# Central factor satisfies image condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., image condition)

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## Contents

## Statement

### Statement with symbols

Suppose is a central factor of a group (in other words, ). Suppose is a surjective homomorphism of groups. Then, is a central factor of .

## Definitions used

### Central factor

`Further information: Central factor`

A subgroup of a group is termed a central factor of , where denotes the centralizer of in .

- Every inner automorphism of restricts to an inner automorphism of .

## Related facts

- Transitive normality satisfies image condition
- SCAB satisfies image condition
- Direct factor satisfies image condition

## Proof

### Proof in terms of centralizers

**Given**: A central factor of a group . A surjective homomorphism .

**To prove**: .

**Proof**: By the definition of homomorphism, if two elements commute in , their images commute in . Thus, the definition of centralizer yields:

.

Taking the product of both sides with yields:

.

By the definition of homomorphism, the left side is the same as , which is (since is a central factor of ). by the assumption of surjectivity, so we get:

.

This forces:

completing the proof.