On the maximum order of torsion elements in \(GL(n,\mathbb{Z})\) and \(\text{Aut}(F_n)\)
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.
scientific article; zbMATH DE number 1230355