(p, q, r) -GENERATIONS OF THE SPORADIC GROUP HN

A finite group G is called (l, m, n) -generated, if it is a quotient group of the triangle group T (l, m, n) = x, y, z | xˡ = yᵐ = zⁿ = xyz = 1. In 16, the question of finding all triples (p, q, r) such that non-abelian finite simple group G is (p, q, r) -generated was posed. In this paper we partially answer this question for the sporadic group HN. In fact, we prove that the sporadic group HN is (p, q, r) -generated if and only if (p, q, r) (2, 3, 5), where p, q and r are prime divisors of |HN| and p q r.