All transpositions involving one element generate the finitary symmetric group
- Transpositions generate the finitary symmetric group
- Transpositions of adjacent elements generate the symmetric group on a finite set
Given: A non-empty set , an element . is the finitary symmetric group on . is the subset of comprising all transpositions involving .
To prove: generates .
- 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 .