J. Appl. Math. & Computing Vol. 10(2002), No. 1 - 2, pp. 167 - 174 GENERATING PAIRS FOR THE HELD GROUP He Ali Reza Ashrafi Abstract. A group G is said to be (l, m, n)-generated if it is a quotient group of the triangle group T (p, q, r)= 〈x, y, z|x p = y q = z r = xyz =1〉. In [15], the question of finding all triples (l, m, n) such that non-abelian finite simple groups are (l, m, n)-generated was posed. In this paper we partially answer this question for the sporadic group He. We continue the study of (p, q, r)−generations of the sporadic simple groups, where p, q, r are distinct primes. The problem is resolved for the Held group He. AMS Mathematics Subject Classification : 20D08, 20F05 Key words and phrases: Held group, (p, q, r)-generation, sporadic group, tri- angle group. 1. Introduction A group G is called (lX, mY, nZ )-generated (or (l, m, n)-generated for short) if there exist x ∈ lX , y ∈ mY and z ∈ nZ such that xy = z and G =< x,y >. If G is (l, m, n)-generated, then we can see that for any permutation π of S 3 , the group G is also ((l)π, (m)π, (n)π)-generated. Therefore we may assume that l ≤ m ≤ n. By [2], if the non-abelian simple group G is (l, m, n)-generated, then either G ∼ = A 5 or 1 l + 1 m + 1 n < 1. Hence for a non-abelian finite simple group G and divisors l, m, n of the order of G such that 1 l + 1 m + 1 n < 1, it is natural to ask if G is a (l, m, n)-generated group. The motivation for this question came from the calculation of the genus of finite simple groups [19]. It can be shown that the problem of finding the genus of a finite simple group can be reduced to one of generations. Moori in [15], posed the problem of finding all triples (l, m, n) such that non- abelian finite simple group G is (l, m, n)-generated. In a series of papers [10- 16], Moori and Ganief established all possible (p, q, r)-generations of the sporadic groups J 1 ,J 2 ,J 3 ,J 4 , HS, McL, Co 2 ,Co 3 , and F 22 , for distinct primes p, q and Received March 7, 2002. Revised June 27, 2002. c  2002 Korean Society for Computational & Applied Mathematics and Korean SIGCAM . 167