# Cocentral implies central factor

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., cocentral subgroup) must also satisfy the second subgroup property (i.e., central factor)

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## Contents

## Statement

Any cocentral subgroup of a group is a central factor.

## Definitions used

### Cocentral subgroup

`Further information: `

A subgroup of a group is a cocentral subgroup if where is the center of .

### Central factor

`Further information: central factor`

A subgroup of a group is a central factor if where is the centralizer of in .

## Related facts

- Cocentrality is upward-closed
- Cocentral implies right-quotient-transitively central factor
- Cocentral implies join-transitively central factor

## Proof

**Given**: A group , a subgroup of such that where is the center of .

**To prove**: where is the centralizer of in .

**Proof**: We have , because any element of the center centralizes every element of . Thus, . Since , we obtain .