# Locally inner automorphism-balanced not implies central factor

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., locally inner automorphism) neednotsatisfy the second subgroup property (i.e., central factor)

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## Contents

## Statement

It is possible to have a group and a locally inner automorphism-balanced subgroup of (i.e., every inner automorphism of restricts to a locally inner automorphism of ) such that is *not* a central factor of .

Here, central factor means that every inner automorphism of restricts to an inner automorphism of . This is equivalent to the condition that .

Note that we cannot have an example where is a finitely generated group (and even more generally, we cannot have an example where is a group with finitely generated inner automorphism group), because finitely generated inner automorphism group implies every locally inner automorphism is inner.

## Facts used

- Finitary symmetric group is locally inner automorphism-balanced in symmetric group
- Finitary symmetric group is centralizer-free in symmetric group
- Restricted direct product is locally inner automorphism-balanced in unrestricted direct product

## Proof

### Proof using the finitary symmetric group

Suppose is an infinite set. Let be the symmetric group on and let be the finitary symmetric group on .

- is locally inner automorphism-balanced in : This follows from Fact (1).
- is not a central factor of : To see this, note that by fact (2), is trivial, so .

### Proof using direct products

Suppose is an infinite collection of (possibly repeated) centerless groups. Let be the external direct product and let be the subgroup of given as the unrestricted direct product.

- is locally inner automorphism-balanced in : This follows from Fact (3).
- is not a central factor of : We can compute that is trivial, and therefore that .