Monomial modular representations and symmetric generation of the Harada-Norton group.
type: Item
instance of: scholarly article
title: Monomial modular representations and symmetric generation of the Harada-Norton group.
zbMATH Open document ID: 1046.20017
DOI: 10.1016/S0021-8693(03)00298-9
published in: Journal of Algebra
publication date: 2003-12-03T00:00:00Z
review text: The second author [in J. Algebra 184, No. 3, 1205-1227 (1996; Zbl 0858.20020)], constructed the Held group \(He\) using a 15-dimensional 7-modular monomial representation of \(3^\cdot A_7\), the triple cover of \(A_7\). He obtained \(He\) as a homomorphic image of a split extension of a free product of 15 cyclic subgroups of order 7 by the group \(3^\cdot A_7\). In this paper, the much larger Harada-Norton group \(HN\) is constructed in a similar manner. A 352-dimensional 5-modular monomial representation of \(2^\cdot HS:2\), a double cover of the automorphism group of the Higman-Sims group \(HS\), is used to build a split extension \(\mathcal P\) of a free product of 352 cyclic subgroups of order 5 by the group \(2^\cdot HS:2\). The group \(\mathcal P\) has \(HN\) as a ``natural'' homomorphic image. An outline of how this approach can be used to construct the 133-dimensional representation of \(HN\) over \(\mathbb{Q}(\sqrt 5)\), which is the smallest degree of a ``true'' characteristic 0 representation of \(\mathcal P\), is given. As in the Held case, extension to the automorphism group of \(HN\) follows easily.
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20C34
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 2012757
zbMATH Keywords: sporadic groups
zbMATH Keywords: symmetric presentations
zbMATH Keywords: modular representations
zbMATH Keywords: matrix group constructions
author: John N. Bray
author: Robert T. Curtis
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
cites work: Q3684278
cites work: Q5289109
cites work: Symmetric generation of the Higman-Sims group
cites work: Monomial modular representations and construction of the Held group
cites work: A Note on the Linear Fractional Group
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description: scientific article; zbMATH DE number 2012757