Short presentations for alternating and symmetric groups

Abstract

We construct two kinds of presentations for the alternating and symmetric groups of degree n n : the first are on two generators in which the number of relations is O ( log ⁡ n ) O(\log n) and the presentation length is O ( log 2 ⁡ n ) O(\log ^2 n) ; the second have a bounded number of generators and relations and length O ( log ⁡ n ) O(\log n) .