Transpositions Generate the Symmetric Group
The set of transpositions generates for every finite .
Theorem 8.11 (Transpositions Generate)
The set of transpositions generates for any finite .
Proof
Induct on . The claim is immediate for . Suppose transpositions generate and take . If , then . Otherwise, for , the permutation fixes and lies in . Thus is a product of transpositions, and so is .
Related
Stated in
- Theorem 8.11 (Transpositions Generate)§8.2 Transpositions and the Sign Homomorphism