# Finitary symmetric group equals center of symmetric group modulo finitary alternating group

From Groupprops

## Contents

## Statement

Let be an infinite set, be the Symmetric group (?) on , be the Finitary symmetric group (?), and be the Finitary alternating group (?). Then, is the Center (?) of .

## Proof

We need to pove that if is such that , then . We do this by picking any , and show that is *not* contained in .

If , must move infinitely many elements. There are two cases, that are collectively exhaustive:

- contains infinitely many finite cycles.
- contains an infinite cycle.

### Case of infinitely many finite cycles

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

### Case of an infinite cycle

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]