# All transpositions involving one element generate the finitary symmetric group

From Groupprops

## Contents

## Statement

Suppose is a non-empty set, an element, and is the Finitary symmetric group (?) on . Then, the set of all Transposition (?)s in involving the element is a generating set for .

When is finite, this generating set is both a Sims-reduced generating set and a Jerrum-reduced generating set.

## Related facts

- Transpositions generate the finitary symmetric group
- Transpositions of adjacent elements generate the symmetric group on a finite set

## Facts used

## Proof

**Given**: A non-empty set , an element . is the finitary symmetric group on . is the subset of comprising all transpositions involving .

**To prove**: generates .

**Proof**:

- Every transposition is in the subgroup generated by : Consider a transposition , with . If either or , . If neither equals , we still have , with all elements on the right side being in . Thus, is in the subgroup generated by .
- The previous step and fact (1) yield that the subgroup generated by is the whole group .