Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On recognition of simple classical groups with prime graph independence number $4$ by spectrum

Maria A. Grechkoseeva, Vladislav M. Rodionov

Abstract

Let $L$ be one of the finite simple classical groups $L_8(q)$, $U_8(q)$, $O_{10}^+(q)$, $O_{10}^-(q)$ or $O_{12}^+(q)$, with $q$ odd. We prove that every finite group having the same set of element orders as $L$ is an almost simple group with socle isomorphic to $L$. This completes the study of the recognition-by-spectrum problem for simple classical groups whose prime graph independence number is equal to $4$.

On recognition of simple classical groups with prime graph independence number $4$ by spectrum

Abstract

Let be one of the finite simple classical groups , , , or , with odd. We prove that every finite group having the same set of element orders as is an almost simple group with socle isomorphic to . This completes the study of the recognition-by-spectrum problem for simple classical groups whose prime graph independence number is equal to .

Paper Structure

This paper contains 6 sections, 33 theorems, 26 equations, 4 tables.

Table of Contents

  1. Notation and some general results
  2. Preliminary calculations and some properties of prime graphs and spectra
  3. Proof of Theorem \ref{['t:main']}: Case $t(S)\neq 4$
  4. General observations
  5. Case $t(S)\geq 5$
  6. Proof of Theorem \ref{['t:main']}: Case $t(S)=4$

Key Result

Theorem 1

Suppose that $L$ is a finite simple classical group over a field of odd characteristic and $t(L)=4$. Then the problem of recognition by spectrum is solved for $L$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (58)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1.1
  • Lemma 1.2: Bang--Zsigmondy
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • ...and 48 more