A note on the product of two permutations of prescribed orders

We prove a conjecture by Stefan Kohl on the existence of triples of permutations of bounded degree with prescribed orders and product 11. More precisely, let a,b,ca,b,c be integers, all ≥2≥2. Then there exist elements x,y,z∈Sc+2x,y,z∈Sc+2 of orders aa, bb and cc respectively, with xyz=1xyz=1. This result leads to an existence result for covers of the complex projective line with bounded degree and prescribed ramification indices.