# Finitary symmetric group is centralizer-free in symmetric group

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Finitary symmetric group (?)) satisfying a particular subgroup property (namely, Centralizer-free subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

## Contents

## Statement

### For the symmetric group, plain

Let be a set with at least three elements. Let denote the symmetric group on (the group of all permutations on ). Then, is trivial.

### For the symmetric group and finitary symmetric group

Let be a set with at least three elements. Let denote the symmetric group on (the group of all permutations on , and denote the subgroup comprising all *finitary* permutations -- permutations moving only finitely many elements. Then the centralizer of is is the trivial group. In other words, no non-identity permutation commutes with every finitary permutation.

### Relation between the two formulations

The second formulation is stronger than the first. For finite sets, the two formulations are precisely the same.

## Related facts

### Stronger facts

### Corollaries

## Proof

**Given**: A set with at least three elements. is the group of all permutations of , and is the subgroup comprising finitary permutations.

**To prove**: is the trivial group.

**Proof**: Suppose and . Our goal is to show that .

Since has at least three elements, we can find a such that and . Let be the transposition . Then, we have:

.

On the other hand, we know that:

.

Thus, the transposition and the transposition are equal. This forces , so either or . The latter case was ruled out by choice of , so , completing the proof.