# Imprimitive Irreducible Modules for Finite Quasisimple Groups

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*Gerhard Hiss; William J. Husen; Kay Magaard*

Motivated by the maximal subgroup problem of the finite
classical groups the authors begin the classification of imprimitive
irreducible modules of finite quasisimple groups over algebraically
closed fields \(K\). A module of a group \(G\) over
\(K\) is imprimitive, if it is induced from a module
of a proper subgroup of \(G\).

The authors obtain their strongest results when \({\rm char}(K)
= 0\), although much of their analysis carries over into positive
characteristic. If \(G\) is a finite quasisimple group of Lie
type, they prove that an imprimitive irreducible \(KG\)-module
is Harish-Chandra induced. This being true for \(\mbox{\rm
char}(K)\) different from the defining characteristic of
\(G\), the authors specialize to the case \({\rm char}(K) = 0\)
and apply Harish-Chandra philosophy to classify irreducible
Harish-Chandra induced modules in terms of Harish-Chandra series, as
well as in terms of Lusztig series. The authors determine the
asymptotic proportion of the irreducible imprimitive
\(KG\)-modules, when \(G\) runs through a series groups
of fixed (twisted) Lie type. One of the surprising outcomes of their
investigations is the fact that these proportions tend to
\(1\), if the Lie rank of the groups tends to infinity.

For exceptional groups \(G\) of Lie type of small rank, and
for sporadic groups \(G\), the authors determine all irreducible
imprimitive \(KG\)-modules for arbitrary characteristic of
\(K\).

#### Table of Contents

# Table of Contents

## Imprimitive Irreducible Modules for Finite Quasisimple Groups

- Cover Cover11 free
- Title page i2 free
- Acknowledgements 18 free
- Chapter 1. Introduction 310
- Chapter 2. Generalities 714
- Chapter 3. Sporadic Groups and the Tits Group 1522
- Chapter 4. Alternating Groups 2532
- Chapter 5. Exceptional Schur Multipliers and Exceptional Isomorphisms 2936
- Chapter 6. Groups of Lie type: Induction from non-parabolic subgroups 4552
- Chapter 7. Groups of Lie type: Induction from parabolic subgroups 8390
- Chapter 8. Groups of Lie type: char(𝐾)=0 9198
- Chapter 9. Classical groups: 𝑐ℎ𝑎𝑟(𝐾)=0 95102
- Chapter 10. Exceptional groups 103110
- Bibliography 109116
- Back Cover Back Cover1126