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Recognition by element orders for simple linear and unitary groups

Maria A. Grechkoseeva, Alexey M. Staroletov, Andrey V. Vasil'ev

Abstract

For a finite group $G$, let $ω(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $ω(H)=ω(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if $h(G)$ is finite, then all finite groups $H$ with $ω(H)=ω(G)$ are described. We complete the solution of the recognition problem for the finite simple linear and unitary groups.

Recognition by element orders for simple linear and unitary groups

Abstract

For a finite group , let be the set of element orders of and let be the number of pairwise nonisomorphic finite groups with . We say that the recognition problem is solved for if the number is known, and if is finite, then all finite groups with are described. We complete the solution of the recognition problem for the finite simple linear and unitary groups.

Paper Structure

This paper contains 5 sections, 31 theorems, 16 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The structure of groups isospectral to classical groups
  4. Proof of Theorems \ref{['t:main']} and \ref{['t:l9l10']}
  5. Proof of Theorem \ref{['t:main2']}

Key Result

Theorem 1

Suppose that $L$ is a finite nonabelian simple linear or unitary group. If $L$ is one of the groups $L_2(9)$, $L_3(3)$, $U_4(2)$, $U_5(2)$, $U_3(5)$, and $U_3(q)$, where $q$ is a Mersenne prime such that $q^2-q+1$ is also a prime, then $h(L)=\infty$. Otherwise, $h(L)<\infty$, every finite group $G$

Theorems & Definitions (59)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Theorem 2
  • Theorem 3
  • Lemma 2.1: Bang--Zsigmondy
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 49 more