# Centerless implies inner automorphism group is centralizer-free in automorphism group

## Contents

## Statement

Suppose is a Centerless group (?): the center of is trivial. Then, the Inner automorphism group (?) (which is naturally isomorphic to ) is a Centralizer-free subgroup (?) inside : there is no non-identity automorphism of that commutes with every inner automorphism.

Equivalently, under the identification of with , we say that is centralizer-free in . Note that this also implies that is a centerless group.

## Related facts

## Facts used

- Group acts as automorphisms by conjugation: In particular, , where is the map .

## Proof

### Hands-on proof

**Given**: A group , with automorphism group and inner automorphism group . Here, for , the inner automorphism is defined as .

**To prove**: If commutes with for all , then is the identity map.

**Proof**:

- For all , : For any , the left side is . Thus, the two sides are equal for all , and are hence equal as functions.
- If and commute, we have : This follows directly from step (1).
- If and commute, we have that is the identity map: Since , we get is the identity map. Using the fact that conjugation is a group action, we obtain that is the identity map.
- (
**Given data used**: is centerless): If and commute, : Step (3) shows that acts trivially by conjugation, and is hence in the center. Since we know that is centerless, must be the identity element, yielding . - If commutes with for every , is the identity map: This is a direct consequence of step (4), applied to all .

### Proof using the commutator language

**Given**: A group , with automorphism group and inner automorphism group . Here, for , the inner automorphism is defined as .

**To prove**: If commutes with for all , then is the identity map.

**Proof**: For convenience, we identify with the subgroup of . Note that now has two interpretations:

- It is the subgroup of generated by commutators between elements of and the automorphism , viewed as
*automorphisms*of . - It is the subgroup of generated by elements of the form , where .

The former interpretation tells us that is trivial, which, viewed using the latter interpretation, yields that is the identity automorphism.