Subgroups of \(\text{GL}_n(q)\) which fix sublattices of the subspace lattice
type: Item
instance of: scholarly article
title: Subgroups of \(\text{GL}_n(q)\) which fix sublattices of the subspace lattice
zbMATH Open document ID: 1018.20044
DOI: 10.1016/S0021-8693(02)00560-4
author: John Shareshian
author: Robert Guralnick
published in: Journal of Algebra
publication date: 2003-04-02T00:00:00Z
review text: Let \(n\) be a positive integer and let \(q=p^e\) be a prime power. An interesting problem of computational group theory is that of producing an algorithm which, given a collection of nonsingular \((n\times n)\)-matrices over the field \(\text{GF}(q)\), returns a useful description of the subgroup of \(\text{GL}_n(q)\) which they generate. A key tool in the algorithms developed so far is Aschbacher's description of the maximal subgroups of \(\text{GL}_n(q)\). That is, \textit{M. Aschbacher} [in Invent. Math. 76, 459-514 (1984; Zbl 0537.20023)] described nine classes \({\mathcal C}_1,\dots,{\mathcal C}_8\), \(\mathcal S\) of proper subgroups of \(\text{GL}_n(q)\) and showed that every proper subgroup of \(\text{GL}_n(q)\) is contained in a group from these nine classes. The subgroups of \(\text{GL}_n(q)\) from classes \({\mathcal C}_1,\dots,{\mathcal C}_8\) are (for the most part) stabilizers in \(\text{GL}_n(q)\) of ``natural'' algebraic structures on the corresponding vector space. If a subgroup \(H\) of \(\text{GL}_n(q)\) is not contained in any member of \(\bigcup_{i=1}^8{\mathcal C}_i\) then \(H/Z(H)\) is almost simple. The group \(G=\text{GL}_n(q)\) acts on the lattice \(B_n(q)\) of all subspaces of the \(n\)-dimensional space \(V=V(n,q)\) over \(\text{GF}(q)\) on which \(G\) acts naturally. For a sublattice \(\mathcal L\) of \(B_n(q)\), by \(G_{\mathcal L}\) is denoted the setwise stabilizer of \(\mathcal L\) in \(G\). Members of certain classes \({\mathcal C}_i\) are stabilizers in \(G\) of certain sublattices of \(B_n(q)\) and the \({\mathcal C}_i\), whose elements are sublattice stabilizers, are known. Recent algorithms (see \textit{C. R. Leedham-Green} and \textit{E. A. O'Brien} [J. Algebra 189, No. 2, 514-528 (1997; Zbl 0884.51006), Int. J. Algebra Comput. 7, No. 5, 541-559 (1997; Zbl 0907.20025)]) involve recognizing \(H\) as being contained in a member of \({\mathcal C}_i\) for certain \(i\) by showing that \(H\) stabilizes some sublattice of \(B_n(q)\)…
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 51D25
zbMATH DE Number: 1891316
zbMATH Keywords: finite linear groups
zbMATH Keywords: Aschbacher classes of subgroups
zbMATH Keywords: subspace lattices
zbMATH Keywords: stabilizers of sublattices
zbMATH Keywords: maximal subgroups
zbMATH Keywords: almost simple groups
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00560-4
OpenAlex ID: W2046537833
description: scientific article; zbMATH DE number 1891316