Conjugate generation of sporadic almost simple groups

TL;DR

This paper determines the minimal number of conjugates of a prime order automorphism needed to generate a subgroup containing a given sporadic simple group, completing the classification for all but the Monster group.

Contribution

It provides a complete calculation of the conjugate generation parameter for all sporadic simple groups except the Monster, advancing understanding of their automorphism structures.

Findings

Calculated conjugate generation numbers for all sporadic groups except the Monster.

Identified automorphisms of prime order that generate subgroups containing the simple group.

Extended previous partial results to a comprehensive classification.

Abstract

As defined by Guralnick and Saxl, given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number $\alpha_S(x)$ is the minimum number of conjugates of $x$ in $\langle x,S\rangle$ that generate a subgroup containing $S$. In this paper, for every sporadic group $S$ other than the Monster and an automorphism $x$ of $S$ of prime order, we complete the determination of the precise value of $\alpha_S(x)$.