Elements with large irreducible submodules contained in maximal subgroups of the general linear group

Sabina Barbara Pannek

Research output: ThesisDoctoral Thesis

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Abstract

We call an element of the finite general linear group GL(V) fat if it leaves invariant and acts irreducibly on a subspace of dimension greater than dim(V)/2. Fat elements can be identified efficiently in practice and are intended to provide the basis for new algorithms dealing with linear groups. We first develop a framework necessary to study fat elements obtaining new results in elementary number theory, theory of finite fields (counting certain irreducible polynomials) and group theory (regarding irreducible semilinear mappings). Then, guided by Aschbacher's theorem, we investigate the occurrence of fat elements in various maximal subgroups of GL(V).
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • The University of Western Australia
  • RWTH Aachen University
Award date21 Jan 2019
DOIs
Publication statusUnpublished - 2019

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Irreducible polynomial
General Linear Group
Maximal Subgroup
Linear Group
Group Theory
Number theory
Semilinear
Galois field
Counting
Subspace
Invariant
Necessary
Theorem
Framework

Cite this

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title = "Elements with large irreducible submodules contained in maximal subgroups of the general linear group",
abstract = "We call an element of the finite general linear group GL(V) fat if it leaves invariant and acts irreducibly on a subspace of dimension greater than dim(V)/2. Fat elements can be identified efficiently in practice and are intended to provide the basis for new algorithms dealing with linear groups. We first develop a framework necessary to study fat elements obtaining new results in elementary number theory, theory of finite fields (counting certain irreducible polynomials) and group theory (regarding irreducible semilinear mappings). Then, guided by Aschbacher's theorem, we investigate the occurrence of fat elements in various maximal subgroups of GL(V).",
keywords = "computational group theory, general linear group, maximal subgroups, irreducible submodules, irreducible polynomials, semilinear mappings, ppd-elements",
author = "Pannek, {Sabina Barbara}",
year = "2019",
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language = "English",
school = "The University of Western Australia, RWTH Aachen University",

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Elements with large irreducible submodules contained in maximal subgroups of the general linear group. / Pannek, Sabina Barbara.

2019.

Research output: ThesisDoctoral Thesis

TY - THES

T1 - Elements with large irreducible submodules contained in maximal subgroups of the general linear group

AU - Pannek, Sabina Barbara

PY - 2019

Y1 - 2019

N2 - We call an element of the finite general linear group GL(V) fat if it leaves invariant and acts irreducibly on a subspace of dimension greater than dim(V)/2. Fat elements can be identified efficiently in practice and are intended to provide the basis for new algorithms dealing with linear groups. We first develop a framework necessary to study fat elements obtaining new results in elementary number theory, theory of finite fields (counting certain irreducible polynomials) and group theory (regarding irreducible semilinear mappings). Then, guided by Aschbacher's theorem, we investigate the occurrence of fat elements in various maximal subgroups of GL(V).

AB - We call an element of the finite general linear group GL(V) fat if it leaves invariant and acts irreducibly on a subspace of dimension greater than dim(V)/2. Fat elements can be identified efficiently in practice and are intended to provide the basis for new algorithms dealing with linear groups. We first develop a framework necessary to study fat elements obtaining new results in elementary number theory, theory of finite fields (counting certain irreducible polynomials) and group theory (regarding irreducible semilinear mappings). Then, guided by Aschbacher's theorem, we investigate the occurrence of fat elements in various maximal subgroups of GL(V).

KW - computational group theory

KW - general linear group

KW - maximal subgroups

KW - irreducible submodules

KW - irreducible polynomials

KW - semilinear mappings

KW - ppd-elements

U2 - 10.26182/5c89d6885683c

DO - 10.26182/5c89d6885683c

M3 - Doctoral Thesis

ER -