Generating symmetric and alternating groups.

We provide three generalizations of all best known generating sets for S n and A n : (1) m -cycles with consecutive entries; (2) m -cycles with fixed first ℓ entries; (3) two cycles, either (1 , ... , m) , (1 , ... , n) or (1 , ... , m) , (2 , ... , n). Our work on (3) fills a gap in the history of the problem, going back to a 1939 paper by S. Piccard [Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116(1) (1939) 752–767]. [ABSTRACT FROM AUTHOR]