# Difference between revisions of "Finitary symmetric group on countable set is subgroup-universal for finite groups"

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(New page: {{embeddability theorem}} ==Statement== Let <math>\Omega</math> be a countably infinite set and <math>G := \operatorname{FSym}(\Omega)</math> be the finitary symmetric group on <math...) |
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Let <math>\Omega</math> be a countably infinite set and <math>G := \operatorname{FSym}(\Omega)</math> be the [[finitary symmetric group]] on <math>\Omega</math>. Then, <math>G</math> is '''subgroup-universal''' for finite groups. In other words, if <math>H</math> is any [[finite group]], <math>H</math> is isomorphic to a subgroup of <math>G</math>. | Let <math>\Omega</math> be a countably infinite set and <math>G := \operatorname{FSym}(\Omega)</math> be the [[finitary symmetric group]] on <math>\Omega</math>. Then, <math>G</math> is '''subgroup-universal''' for finite groups. In other words, if <math>H</math> is any [[finite group]], <math>H</math> is isomorphic to a subgroup of <math>G</math>. | ||

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+ | ==Related facts== | ||

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+ | * [[Free group on countable set is quotient-universal for finitely generated groups]] | ||

+ | * [[Free group on two generators is SQ-universal]] | ||

==Facts used== | ==Facts used== |

## Latest revision as of 17:14, 14 October 2008

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.