Alternating groups, Hurwitz groups and \(H^*\)-groups. (Q1770505)

The authors obtain the pairs of generators, necessary to study the non-orientable case, of the alternating groups \(A_n\) for \(n=15\), 21, 22, 28 and 29, which are also Hurwitz groups, groups with maximal number of automorphisms on Riemann surfaces. The results found here can be applied to handle the corresponding problem on non-orientable surfaces. In particular, they show that the ones for \(n=15\) and 28 match the bound for non-orientable surfaces, while the ones for \(n=21\), 22 and 29 do not. They also obtain some other Hurwitz groups which are at the same time proper subgroups of the alternating groups. They obtain a way of deciding which alternating groups are also \(H^*\)-groups.