All transpositions involving one element generate the finitary symmetric group

Statement

Suppose $S$ is a non-empty set, $s\in S$ an element, and $G=\operatorname {FSym} (S)$ is the Finitary symmetric group (?) on $S$. Then, the set of all Transposition (?)s in $S$ involving the element $s$ is a generating set for $G$.

When $S$ 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

1. Transpositions generate the finitary symmetric group

Proof

Given: A non-empty set $S$, an element $s\in S$. $G$ is the finitary symmetric group on $S$. $T$ is the subset of $G$ comprising all transpositions involving $S$.

To prove: $T$ generates $G$.

Proof:

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