All transpositions involving one element generate the finitary symmetric group

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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

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.