Alternating groups, Hurwitz groups and [formula omitted]-groups

Abstract

Riemann surfaces of genus g admit at most 84 ( g − 1 ) automorphisms. The group attaining this bound is called a Hurwitz group. A group G is a Hurwitz group if and only if it has a pair of generators of order 2 and 3 whose product has order 7. Each alternating group A n for n > 168 is a Hurwitz group and most cases with n < 168 are too, but the suitable generators are not explicitly known. In this paper we obtain all pairs of such generators of Hurwitz groups A n for n < 35 , namely A 15 , A 21 , A 22 , A 28 , A 29 . These results are used to deal with the corresponding problem on non-orientable surfaces. In this case the question is stated in terms of finding a third element of order 2 whose products with the previous elements have also order two. In particular, we obtain that A 15 and A 28 match the bound for non-orientable surfaces (that is to say they are H * -groups) whilst A 21 , A 22 , and A 29 do not. As a byproduct we obtain other Hurwitz groups which are proper subgroups of A n and give some examples of such generators for them.