C-groups of Suzuki type
type: Item
instance of: scholarly article
title: C-groups of Suzuki type
zbMATH Open document ID: 1330.51014
author: Thomas Connor
author: Dimitri Leemans
published in: Journal of Algebraic Combinatorics
publication date: 2015-11-04T00:00:00Z
review text: The non-abelian simple groups are generated by their involutions, and so are likely places to look for abstract regular polytopes (which are certain thin geometries), whose automorphism groups are string C-groups. A C-group \((G,\{\rho_0,\ldots,\rho_{r-1}\})\) has involutory generators \(\rho_i\) which satisfy the intersection property \[ \langle \rho_i \mid i \in I \rangle \cap \langle \rho_j \mid i \in J \rangle = \langle \rho_k \mid k \in I \cap J \rangle \] for all \(I,J \subseteq \{0,\ldots,r-1\}\); the string condition says that \(\rho_i\rho_j = \rho_j\rho_i\) whenever \(|i - j| > 1\) (possibly after reordering of the \(\rho_i\)). A group is nearly simple if it lies between a simple group and its automorphism group. After a discussion of the representation of dihedral groups as C-groups and the structure of Suzuki groups, the authors prove the following. Let \(q = 2^{2e+1} \neq 2\), and let \(\mathrm{Sz}(q) \leq G \leq \mathrm{Aut}(\mathrm{Sz}(q))\) be an almost simple group of Suzuki type. If \((G,\{\rho_0,\ldots,\rho_{r-1}\})\) is a C-group, then \(G = \mathrm{Sz}(q)\) and \(r = 3\). Moreover, there exist at least one such string C-group \(G\) and at least one non-C-group \(G\).
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 6504087
zbMATH Keywords: Suzuki groups
zbMATH Keywords: C-groups
zbMATH Keywords: thin geometries
zbMATH Keywords: abstract regular polytopes
reviewed by: Peter McMullen
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s10801-015-0605-2
OpenAlex ID: W2161861503
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DOI: 10.1007/S10801-015-0605-2
description: scientific article; zbMATH DE number 6504087