# Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible

From Groupprops

## Contents

## Statement

Suppose is a group with center , inner automorphism group , and automorphism group . Suppose the following two statements are true:

- Every automorphism of is a center-fixing automorphism: it fixes every element of the center .
- is a maximal subgroup of .

Then, every automorphism of is a normal-extensible automorphism: whenever is a normal subgroup of some group and is an automorphism of , extends to an automorphism of .

## Related facts

- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
- Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensible
- Every automorphism is center-fixing and outer automorphism group is rank one p-group implies not every normal-extensible automorphism is inner

## Facts used

- Center-fixing implies central factor-extensible: If is a center-fixing automorphism of a central factor of a group , extends to an automorphism of .
- Equivalence of definitions of central factor

## Proof

**Given**: A group such that every automorphism of fixes every element of , and is a maximal subgroup of .

**To prove**: For any automorphism of , and any group containing as a normal subgroup, extends to an automorphism of .

**Proof**: Let be the homomorphism given by the action of on by conjugation. This map exists because is normal in .

- The image of is either or : The image of is a subgroup of . It contains , because equals . Since is a maximal subgroup of , the image is either or .
- If the image is , then every automorphism of extends to an inner automorphism of , and we are done. (We say in this case that is a normal fully normalized subgroup of ).
- If the image is , then is a central factor of , i.e., (see fact (2)). Thus, by fact (1), any automorphism that fixes the center of extends to an automorphism of . Since we assumed that
*every*automorphism fixes the center of , we obtain that every automorphism of extends to an automorphism of .