On the Exact Spread of Sporadic Simple Groups

Abstract

A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold (1975a) defined a finite group G to have spread r if for every set {x ₁, x ₂,…, x _r } of distinct nontrivial elements of G, there exists an element y ∊ G such that G = 〈 x _i , y〉 for all i. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief (1996) in his Ph.D. thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In Ganief and Moori (2001) the second author and Ganief used probabilistic methods and established a reasonable lower bound for the exact spread s(G) of each sporadic simple group G. The present article deals with the search for reasonable upper bounds for the exact spread of the sporadic simple groups.

Key Words:

• Exact spreads
• Sporadic

2000 Mathematics Subject Classification:

• 20D08
• 20F05

ACKNOWLEDGMENT

The authors express their gratitude to the referee whose suggestions and comments led to significant improvement of the content and the presentation of this article. They also would like to thank Dr. P. E. Holmes who provided the relevant information on maximal subgroups of the Monster.

Notes

Communicated by M. R. Dixon.