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Refined conjugate generation in sporadic groups

Danila O. Revin, Andrei V. Zavarnitsine

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 $4$.

Refined conjugate generation in sporadic groups

Abstract

Given an automorphism of order bigger than of a sporadic simple group , we show that there are at most conjugates of required to generate a subgroup of order divisible by a fixed prime divisor of . The only exception is the case where , is in class , , and then the required number of generators is .
Paper Structure (4 sections, 7 theorems, 20 equations, 2 tables)

This paper contains 4 sections, 7 theorems, 20 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of main result
  4. Final remarks and open problems

Key Result

Theorem 1

Let $S$ be a sporadic group, let $x\in \operatorname{Aut} S$ with $|x| > 2$, and let $r$ be a prime divisor of $|S|$ coprime to $|x|$. Then $2\leqslant \beta_{r,S}(x)\leqslant 3$, except when $(S,x^S,r)=(Suz,3A,11)$ and then $\beta_{r,S}(x)=4$.

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Proposition 1
  • Proposition 2: Scott
  • Corollary 2
  • proof
  • Proposition 3
  • Lemma 1
  • proof
  • proof