New Upper Bounds on the Spreads of Some Large Sporadic Groups

TL;DR

This paper establishes improved upper bounds on the spread of various large sporadic simple groups, significantly advancing understanding of their generative properties.

Contribution

It provides the first substantial bounds on the spread of many sporadic simple groups, improving previous results by several orders of magnitude.

Findings

New upper bounds for the spread of multiple sporadic groups

Significant reduction in known bounds for these groups

Enhanced understanding of the generative properties of sporadic groups

Abstract

Let G be a group. We say that G has spread r if for any set of distinct elements {x1,..., xr}\subset G there exists an element y\in G with the property that <xi, y>=G for every 0<i<r+1. Few bounds on the spread of finite simple groups are known. In this paper we present improved upper bounds for the spread of many of the sporadic simple groups, in some cases improving on the best known upper bound by several orders of magnitude.