Maximal Solvable Subgroups of Finite Classical Groups

Preface
Contents
1 Introduction
	1.1 Introduction and Historical Background
	1.2 Basic Notation and Terminology
	1.3 Reduction to Linear Groups
2 Basic Structure of Maximal Irreducible Solvable Subgroups
	2.1 Construction of Maximal Irreducible Solvable Groups
		2.1.1 General Setup and Notation
		2.1.2 Generalities on Maximal Solvable Subgroups
		2.1.3 Representation Theory
		2.1.4 Number Theory
	2.2 Metrically Imprimitive Subgroups
	2.3 Metrically Completely Reducible Subgroups
	2.4 Metrically Primitive Maximal Irreducible Solvable Subgroups
	2.5 Groups of Type B_0
	2.6 Groups of Type B_1
	2.7 Groups of Type B_2
	2.8 Groups of Type B_3
	2.9 Groups of Type B_i with μ = 1
3 Extraspecial Groups
	3.1 Extraspecial Groups
	3.2 Absolutely Irreducible Representations of Extraspecial Groups
		3.2.1 Absolutely Irreducible Representation of r+1+2
		3.2.2 Unitary Representation of r+1+2
		3.2.3 Orthogonal Representation of 2+1+2
		3.2.4 Absolutely Irreducible Representation of 2-1+2
		3.2.5 Unitary Representation of 2-1+2
		3.2.6 Symplectic Representation of 2-1+2
4 Metrically Primitive Maximal Irreducible Solvable Subgroups
	4.1 The Fitting Subgroup of C_G(F_0)
	4.2 Structure of Metrically Primitive Maximal Solvable Subgroups
	4.3 Description of Gμ,νB(X1, …, Xk)
		4.3.1 Groups of Type B0
		4.3.2 Groups of Type B1
		4.3.3 Groups of Type B2
		4.3.4 Groups of Type B3
		4.3.5 General Properties of Gμ,νB(X1, …, Xk)
	4.4 Maximal Irreducible Solvable Subgroups of GO(n,q) for n Odd
5 Basic Properties of Gμ,νB(X1, …, Xk)
	5.1 Maximal Irreducible Solvable Subgroups of Multiplier 2
	5.2 Irreducibility of Gμ,νB(X1, …, Xk)
	5.3 Properties of F_0 and A in Gμ,νB(X1, …, Xk)
	5.4 Examples of the Construction in Some Special Cases
	5.5 Some Examples Where Gμ,νB(X1, …, Xk) Is Not Maximal Solvable
	5.6 Primitivity of Gμ,νB(X1, …, Xk)
	5.7 Uniqueness of F_0 and A
	5.8 Conjugacy of Gμ,νB(X1, …, Xk)
6 Fixed Point Spaces and Abelian Subgroups
	6.1 Fixed Point Spaces in Symplectic-Type Normalizers
	6.2 Fixed Point Spaces in Primitive Solvable Linear Groups
	6.3 Abelian Subgroups of Solvable Affine Groups
7 Maximality of the Groups Constructed
	7.1 Maximality of G(X_1, ..., X_k)
	7.2 Completely Reducible Subgroups of G(X_1, …, X_k)
	7.3 Systems of Imprimitivity for Completely Reducible Subgroups
	7.4 Maximality of Metrically Imprimitive Subgroups
	7.5 Maximality of Metrically Completely Reducible Subgroups
	7.6 Further Results
8 Examples
	8.1 Examples and Summary of Construction
	8.2 Tables of Examples
References
Index