On the maximum order of torsion elements in \(GL(n,\mathbb{Z})\) and \(\text{Aut}(F_n)\)
type: Item
instance of: scholarly article
title: On the maximum order of torsion elements in \(GL(n,\mathbb{Z})\) and \(\text{Aut}(F_n)\)
zbMATH Open document ID: 0930.11070
author: Gilbert Levitt
author: Jean-Louis Nicolas
published in: Journal of Algebra
publication date: 2000-02-16T00:00:00Z
review text: Let \(g(n)\) denote the maximal order of an element of the symmetric group on \(n\) letters. \textit{E. Landau} [Arch. Math. Phys. (3) 5, 92-103 (1903; JFM 34.0233.02)] proved that \(\log g(n)\sim(n\log n)^{1/2}\) as \(n\to\infty\). For improved estimates see \textit{J.-P. Massias}, \textit{J.-L. Nicolas} and \textit{G. Robin} [Acta Arith. 50, 221-242 (1988; Zbl 0588.10049)]. As to the number \(\nu(g(n))\) of distinct prime divisors of \(g(n)\) and the largest prime divisor \(P(g(n))\) of \(g(n)\), \textit{J.-L. Nicolas} [Bull. Soc. Math. Fr. 97, 129-191 (1969; Zbl 0184.07202); Acta Arith. 14, 315-332 (1968; Zbl 0179.34804)] showed that \(\nu(g(n))\sim 2(n/\log n)^{1/2}\) and \(P(g(n))\sim(n\log n)^{1/2}\) as \(n\to\infty\). In the paper under review the authors study the maximal order \(G(n)\) and \(H(n)\) of torsion elements in \(\text{GL}(n,\mathbb{Z})\) and the automorphism group \(\Aut(F_n)\) of a free group of rank \(n\), respectively. They prove that \[ \log G(n)\sim(n\log n)^{1/2},\quad\nu\bigl(G(n)\bigr)\sim 2(n/\log n)^{1/2},\quad P\bigl(G(n)\bigr)\sim(n\log n)^{1/2} \] and, for a fixed prime \(p\), the logarithm of the maximal power of \(p\) dividing \(G(n)\) is \(\sim 2^{-1}\log n\), as \(n\to\infty\). Further, \(H(n)=G(n)\) if and only if \(n\neq 2\), 6, 12.
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 1230355
zbMATH Keywords: general linear group
zbMATH Keywords: automorphism group of a free group
zbMATH Keywords: maximal order
zbMATH Keywords: torsion elements
reviewed by: Mihály Szalay
MaRDI profile type: Publication
full work available at URL: jabr.1998.7481
OpenAlex ID: W1967107537
cites work: The centre of groups with one defining relator
cites work: The Largest Prime Dividing the Maximal Order of an Element of S n
cites work: On the orders of finite subgroups of \(\text{GL}(n,\mathbb{Z})\)
cites work: Finite groups of automorphisms of free groups
cites work: Finite subgroups of groups of outer automorphisms of free groups
cites work: Évaluation asymptotique de l'ordre maximum d'un élément du groupe symétrique
cites work: The Maximum Order of an Element of a Finite Symmetric Group
cites work: Sur l'ordre maximum d'un élément dans le groupe $S_n$ des permutations
cites work: Q5580252
cites work: The maximum order of finite groups of outer automorphisms of free groups
DOI: 10.1006/JABR.1998.7481
description: scientific article; zbMATH DE number 1230355