Linear groups of small degree over fields of finite characteristic
type: Item
instance of: scholarly article
title: Linear groups of small degree over fields of finite characteristic
zbMATH Open document ID: 0857.20029
author: Harvey I. Blau
author: Jiping Zhang
published in: Journal of Algebra
publication date: 1997-03-03T00:00:00Z
review text: Finite linear groups over a field \(F\) of characteristic \(p>0\) which have degree less than \(p\) and are generated by their \(p\)-elements are classified. For each of these groups it is pointed out whether it can be lifted to characteristic 0. The whole list is too large to be given here. We shall state the answer for irreducible quasisimple groups only. Let \(G\) be a finite group, \(S=G/Z(G)\) be simple, \(V\) be a faithful irreducible \(G\)-module over \(F\) and \(\dim V<p\). Then one of the following holds: 1) \(\dim V=p-1\) and the pair \((S,p)\) appears in List 1; 2) \(\dim V=p-2\) and the pair \((S,p)\) appears in List 2; 3) \(p=7\), \(S=A_7\), \(\dim V=4\); 4) \(p=11\), \(S=J_1\), \(\dim V=7\); 5) \(S\) is a group of Lie type defined over a field of characteristic \(p\). List 1, 1) \(S=\text{PSp}(2n,q)\), \(n=2^a\), \(a\geq 0\), \(q\) is odd, \(p=(q^n+1)/2\); 2) \(S=\text{PSU}(n,q^2)\) (\(n\) is an odd prime, \(n\nmid(q+1)\)), \(p=(q^n+1)/(q+1)\); 3) \(S=A_5\), \(p=3\); 4) \(S=A_7\), \(p=5\); 5) \(S=\text{PSL}(3,4)\), \(\text{PSU}(4,9)\), \(J_2\), or \(A_7\), \(p=7\); 6) \(S=M_{11}\), \(M_{12}\), or \(M_{22}\), \(p=11\); 7) \(S=Suz\) or \(G_2(4)\), \(p=13\); 8) \(S=J_3\), \(p=19\); 9) \(S=Ru\), \(p=29\). List 2, 1) \(S=A_n\), \(n\) is prime, \(n\geq7\), \(p=n\); 2) \(S=\text{PSL}(n,q)\), \(n\) is prime, \(n\nmid(q-1)\), \(p=(q^n-1)/(q-1)\); 3) \(S=A_6\) or \(A_7\), \(p=5\); 4) \(S=M_{11}\), \(p=11\); 5) \(S=M_{23}\), \(p=23\). For the complex field the problem was solved by the second author [in: Contemp. Math. 82, 243-254 (1989; Zbl 0671.20032)]. The first author [J. Algebra 131, No. 1, 92-109 (1990; Zbl 0696.20013)] classified the linear groups in characteristic \(p\) with a cyclic \(p\)-Sylow subgroup \(P\) and degree \(d\leq|P|-2\), the second one [Proc. Am. Math. Soc. 110, No. 1, 53-57 (1990; Zbl 0724.20031)] did the same for \(d<p-2\), under the assumption that \(P\) is Abelian. The classification of finite simple groups and results of the first aut…
reviewed by: I. D. Suprunenko
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 571351
zbMATH Keywords: finite linear groups
zbMATH Keywords: irreducible quasisimple groups
zbMATH Keywords: faithful irreducible modules
zbMATH Keywords: groups of Lie type
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: classification of finite simple groups
MaRDI profile type: Publication
full work available at URL: jabr.1993.1162
OpenAlex ID: W1998877703
DOI: 10.1006/JABR.1993.1162
Recommended article: A group theoretic characterization of linear groups
Recommended article: Minimal Polynomials and Eigenvalues of p -Elements in Representations of Quasi-Simple Groups with a Cyclic Sylow p -Subgroup
Recommended article: On linear groups of degrees at most \(|{} P|{} -1\)
Recommended article: On real and rational representations of finite groups
Recommended article: Irreducible tensor products of representations of finite quasi-simple groups of Lie type
Recommended article: Minimal modular character degrees for groups with a cyclic Sylow subgroup
Recommended article: Generating Finite Completely Reducible Linear Groups
Recommended article: Q3767484
Recommended article: Quadratic pairs.
Recommended article: Almost simple groups with long orbits in absolutely irreducible action.
description: scientific article; zbMATH DE number 571351