The exact spread of M12 is 9

TL;DR

This paper determines the exact spread of the Mathieu group M12, a finite simple sporadic group, establishing its precise value for the first time in this context.

Contribution

It provides the first known exact spread value for the Mathieu group M12, advancing understanding of spread in finite simple groups.

Findings

The exact spread of M12 is 9.

M12 has spread 9 but not 10.

This result fills a gap in the classification of spreads for sporadic groups.

Abstract

Let G be a group. We say that G has spread r if for any set of distinct non-trivial elements {x1,...,xr}\subset G there exists an element y\in G with the property that <xi, y> = G for every 1 0<i<r+1. The group G has exact spread r if it has spread r but not r + 1. The case where G is a finite simple group is particularly interesting since it is known that in this case the spread is at least 2. The precise value of the exact spread of a simple group is known in very few cases. Here we determine the precise value of the exact spread in the smallest sporadic group for which this is still unknown, the Mathieu group M12.