Book contents
- Frontmatter
- Contents
- Introduction
- 1 Groups and semigroups: connections and contrasts
- 2 Toward the classification of s-arc transitive graphs
- 3 Non-cancellation group computation for some finitely generated nilpotent groups
- 4 Permutation and quasi-permutation representations of the Chevalley groups
- 5 The shape of solvable groups with odd order
- 6 Embedding in finitely presented lattice-ordered groups: explicit presentations for constructions
- 7 A note on abelian subgroups of p-groups
- 8 On kernel flatness
- 9 On proofs in finitely presented groups
- 10 Computing with 4-Engel groups
- 11 On the size of the commutator subgroup in finite groups
- 12 Groups of infinite matrices
- 13 Triply factorised groups and nearrings
- 14 On the space of cyclic trigonal Riemann surfaces of genus 4
- 15 On simple Kn-groups for n = 5, 6
- 16 Products of Sylow subgroups and the solvable radical
- 17 On commutators in groups
- 18 Inequalities for the Baer invariant of finite groups
- 19 Automorphisms with centralizers of small rank
- 20 2-signalizers and normalizers of Sylow 2-subgroups in finite simple groups
- 21 On properties of abnormal and pronormal subgroups in some infinite groups
- 22 P-localizing group extensions
- 23 On the n-covers of exceptional groups of Lie type
- 24 Positively discriminating groups
- 25 Automorphism groups of some chemical graphs
- 26 On c-normal subgroups of some classes of finite groups
- 27 Fong characters and their fields of values
- 28 Arithmetical properties of finite groups
- 29 On prefrattini subgroups of finite groups: a survey
- 30 Frattini extensions and class field theory
- 31 The nilpotency class of groups with fixed point free automorphisms of prime order
4 - Permutation and quasi-permutation representations of the Chevalley groups
Published online by Cambridge University Press: 20 April 2010
- Frontmatter
- Contents
- Introduction
- 1 Groups and semigroups: connections and contrasts
- 2 Toward the classification of s-arc transitive graphs
- 3 Non-cancellation group computation for some finitely generated nilpotent groups
- 4 Permutation and quasi-permutation representations of the Chevalley groups
- 5 The shape of solvable groups with odd order
- 6 Embedding in finitely presented lattice-ordered groups: explicit presentations for constructions
- 7 A note on abelian subgroups of p-groups
- 8 On kernel flatness
- 9 On proofs in finitely presented groups
- 10 Computing with 4-Engel groups
- 11 On the size of the commutator subgroup in finite groups
- 12 Groups of infinite matrices
- 13 Triply factorised groups and nearrings
- 14 On the space of cyclic trigonal Riemann surfaces of genus 4
- 15 On simple Kn-groups for n = 5, 6
- 16 Products of Sylow subgroups and the solvable radical
- 17 On commutators in groups
- 18 Inequalities for the Baer invariant of finite groups
- 19 Automorphisms with centralizers of small rank
- 20 2-signalizers and normalizers of Sylow 2-subgroups in finite simple groups
- 21 On properties of abnormal and pronormal subgroups in some infinite groups
- 22 P-localizing group extensions
- 23 On the n-covers of exceptional groups of Lie type
- 24 Positively discriminating groups
- 25 Automorphism groups of some chemical graphs
- 26 On c-normal subgroups of some classes of finite groups
- 27 Fong characters and their fields of values
- 28 Arithmetical properties of finite groups
- 29 On prefrattini subgroups of finite groups: a survey
- 30 Frattini extensions and class field theory
- 31 The nilpotency class of groups with fixed point free automorphisms of prime order
Summary
Abstract
A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rationals and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the group G2(qn), q ≠ 3.
AMS Classification: 20C15
Keywords: General linear group, Quasi-permutation.
Introduction
Let G be a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space. We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x and so is a non-negative integer.
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- Groups St Andrews 2005 , pp. 424 - 431Publisher: Cambridge University PressPrint publication year: 2007