On the ranks of O’N and Ly

Abstract

Let G be a finite group and X a conjugacy class of G. We denote rank(G : X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks for two sporadic simple groups, namely the O'Nan sporadic simple group O'N and the Lyons sporadic simple group, Ly. Computations were carried with the aid of the computer algebra system GAP [The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.3, Aachen, St Andrews, 2003, http://www.gap-system.org ].

On the ranks ofFi22

“…It has 65 conjugacy classes of its elements including three classes of involutions, namely 2A, 2B and 2C. For basic properties of F i 22 and computational techniques, the reader is encouraged to consult [1,2,3,4,6], [8,13,15] and [24]. Theorem 6.…”

“…This paper is intended as a sequel to the author's earlier work on the determination of ranks for the sporadic simple groups. In a series of papers [1,2,3,4,5,7,6], the author with others have studied the ranks for the sporadic simple groups HS, M cL, J 1 , J 2 , J 3 , J 4 , He, Co 1 , Co 2 , Co 3 , O ′ N and Ly. The motivation for this study is outlined in these papers.…”

On the generation of sporadic simple group O'N by (2,3,t) generators

“…The problem of generating a group by a set of conjugate involutions of minimal size is also closely related to the (2, 3, t)-generation of the group. In a series of articles [1,2,3,4,5], it has been shown that sporadic groups HS,…”

“…Drafsheh, Ashrafi and Moghani [14] computed all possible (p, q, r)-generations of the sporadic simple group O N, where p, q and r are prime divisors of |O N|. Recently, Ali [1] computed ranks (minimal conjugate generating sets) for the O'Nan's sporadic simple group O N. More recently, the author in [6,7] and with Ali [8,9] determined (2, 3, t)-generations for sporadic groups HS, McL, J 1 , J 2 , Co 2 and Co 3 .…”

“…We will also give the generating triples for Held group He. For more information regarding the study of (2, 3, t)-generations and computational techniques used in this article to determine the generating pairs, we refer the reader to the references cited above [1], [4], [15], [18] and [23].…”