Generating Sets of Mathieu Groups

TL;DR

This paper determines the maximum size of irredundant generating sets for the Mathieu groups M11 and M12, using subgroup analysis and computational methods, establishing that these sizes are 5 and 6 respectively.

Contribution

It extends the understanding of generating sets to Mathieu groups by computing exact maximum sizes using subgroup analysis and computational verification.

Findings

Maximum irredundant generating set size for M11 is 5

Maximum irredundant generating set size for M12 is 6

Computational methods confirmed the theoretical bounds

Abstract

Julius Whiston calculated the maximum size of an irredundant generating set for $S_n$ and $A_n$ by examination of maximal subgroups. Using analogous considerations, we will compute upper bounds to this value for the first two Mathieu groups, $M_{11}$ and $M_{12}$. Computational results gave explicit irredundant generating sets of $M_{11}$ and $M_{12}$ of size 5 and 6, respectively. Together these give the full results that the maximum size of an irredundant generating set for $M_{11}$ is 5 and for $M_{12}$ it is 6.