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 fact about::finitary symmetric group on $$S$$. Then, the set of all fact about::transpositions 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) uses::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 $$G$$.