Refined conjugate generation in sporadic groups

TL;DR

This paper investigates the minimal number of conjugates of an automorphism needed to generate subgroups with order divisible by a fixed prime in sporadic simple groups, revealing a mostly uniform bound with a specific exception.

Contribution

It establishes a bound of at most three conjugates for generating such subgroups in most cases, with a detailed exception for the Suzuki group and specific automorphism class.

Findings

At most 3 conjugates generate the subgroup in most cases.

An exception occurs for the Suzuki group with automorphism class 3A and prime 11, requiring 4 conjugates.

The result provides insight into the structure and automorphisms of sporadic groups.

Abstract

Given an automorphism $x$ of order bigger than $2$ of a sporadic simple group $S$, we show that there are at most $3$ conjugates of $x$ required to generate a subgroup of order divisible by a fixed prime divisor $r$ of $|S|$. The only exception is the case where $S=Suz$, $x$ is in class $3A$, $r=11$, and then the required number of generators is $4$.