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

Abstract

A finite group $G$ is called $(l,m,n)$-generated, if it is a quotient group of the triangle group $T(l,m,n) = \langle x,y,z | x^l = y^m = z^n = xyz = 1 \rangle$.

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 $(p,q,r) \ne (2,3,5)$, where $p, q$ and $r$ are prime divisors of $|HN|$ and $p \lt q \lt r$.