Bounding the order of complex linear groups and permutation groups with selected composition factors
type: Item
instance of: scholarly article
title: Bounding the order of complex linear groups and permutation groups with selected composition factors
zbMATH Open document ID: 1543.20007
author: Geoffrey R. Robinson
published in: Transactions of the American Mathematical Society
publication date: 2024-07-04T00:00:00Z
review text: In this interesting article, the author gets some general bounds on the order of finite subgroups of \(\mathrm{GL}(n, \mathbb{C})\). The work is originally motivated by some question of P. Etingof (related to growth rates of tensor powers in symmetric tensor categories, see [\textit{K. Coulembier} et al., Pure Appl. Math. Q. 20, No. 3, 1141--1179 (2024; Zbl 1537.18020)]). The generality of the methods used lead the reviewer to provide a description of the main concepts developed and the main results obtained by the author.\N\NLet \(G\) be a finite group. A section of \(G\) is a group of the form \(H/K\) where \(H \leq G\) and \(K \trianglelefteq H\), a section \(G/H\) is normal if \(H \trianglelefteq G\). Let \(\mathbf{P}\) be a property of finite groups which is inherited by all sections (hereditary for brevity) and we let \(\mathbf{Q}\) be a property which is inherited by all normal sections (normally hereditary for brevity). A normally hereditary property \(\mathbf{Q}\) is extant if at least one finite group has property \(\mathbf{Q}\) and an extant property \(\mathbf{Q}\) is irredundant if there is at least one finite group which does not have property \(\mathbf{Q}\). If \(\mathbf{Q}\) is an extant normally hereditary property of finite groups and \(G\) is finite let\N\(A(\mathbf{Q})= \{ |G : F(G)|^{1/(n-1)} \mid n \geq 2, G \leq \mathrm{GL}(n,\mathbb{C}) \text{ and } G \text{ has } \mathbf{Q} \} \),\N\(B(\mathbf{Q})= \{ |G|^{1/(n-1)} \mid n \geq 2, G \leq \mathrm{Sym}(n) \text{ and } G \text{ has } \mathbf{Q} \} \),\N\(C(\mathbf{Q})= \{ |G : A|^{1/(n-1)} \mid n \geq 2, G \leq \mathrm{GL}(n,\mathbb{C}) \text{ and } G \text{ has } \mathbf{Q} \} \),\Nwhere \(A\) is a maximal normal subgroup of the finite group \(G\).\N\NThe first general result, which makes use of the \textsf{CFSG}, is Theorem 1.1: Let \(\mathbf{P}\) be an extant irredundant hereditary property of finite groups. Then the sets \(A(\mathbf{P})\), \(B(\mathbf{P})\) and \(C(\mathbf{P})\) are all bound…
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 7876018
zbMATH Keywords: subgroup of complex linear group
zbMATH Keywords: subgroup of symmetric group
zbMATH Keywords: bound for the order
zbMATH Keywords: composition factor
MaRDI profile type: Publication
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DOI: 10.1090/TRAN/9091
description: scientific article; zbMATH DE number 7876018