# Finitary symmetric group on countable set is subgroup-universal for finite groups

From Groupprops

This article gives the statement, and possibly proof, of an embeddability theorem: a result that states that any group of a certain kind can be embedded in a group of a more restricted kind.

View a complete list of embeddability theorems

## Contents

## Statement

Let be a countably infinite set and be the finitary symmetric group on . Then, is **subgroup-universal** for finite groups. In other words, if is any finite group, is isomorphic to a subgroup of .

## Related facts

- Free group on countable set is quotient-universal for finitely generated groups
- Free group on two generators is SQ-universal

## Facts used

- Cayley's theorem: This states that every group embeds as a subgroup of the symmetric group on its underlying set.

## Proof

The proof follows from fact (1), and the observation that the symmetric group on any finite set is isomorphic to some subgroup of -- namely, the subgroup comprising the permutations on a finite subset of of the same cardinality.