# Central factor-extensible automorphism

From Groupprops

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This is a variation of extensible automorphism|Find other variations of extensible automorphism |

## Definition

### Symbol-free definition

An automorphism of a group is termed **central factor-extensible** if, for every embedding of the group as a central factor of a group, the automorphism can be extended to an automorphism of the bigger group.

### Definition with symbols

An automorphism of a group is termed **central factor-extensible** if, for every embedding of as a central factor in a group , there exists an automorphism of whose restriction to is .

## Relation with other properties

### Stronger properties

- Extensible automorphism
- Normal-extensible automorphism
- Center-fixing automorphism:
`For full proof, refer: Center-fixing implies central factor-extensible`

## Facts

In a centerless group, *every* automorphism is central factor-extensible. This is because any central factor that is centerless as a group must be a direct factor.

`Further information: Centerless implies every automorphism is central factor-extensible`