Computational Abstract Algebra: Using Monomial Matrices to Represent Groups in GAP

Persistent Link:
http://hdl.handle.net/10150/244772
Title:
Computational Abstract Algebra: Using Monomial Matrices to Represent Groups in GAP
Author:
Rome, Zachary Robert
Issue Date:
May-2012
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
A monomial matrix is a matrix with exactly one non-zero element in each row and column. We will utilize GAP to construct all (transitive) representations of a given group using monomial matrices. First, essential group theory definitions and theorems will be provided, as well as an in-depth look at table of marks and monomial matrices. After describing the necessary mathematics, we will explore the GAP programming needed to achieve this goal. Ultimately we want a table, where each row represents a subgroup of the given group and, within the row, the table will hold the linear characters fixed by the monomial matrices of that subgroup. We will furthermore explore how to represent monomial matrices computationally in different ways and how to create a data structure to represent them. Our final goal will require GAP functions for finding all homomorphisms from a subgroup to roots of unity, using these homomorphisms to create monomial matrix representations of the group, and iterating through the subgroups of the group (up to conjugacy) to find all (transitive) monomial matrix representations of the group.
Type:
text; Electronic Thesis
Degree Name:
B.S.
Degree Level:
bachelors
Degree Program:
Honors College; Mathematics
Degree Grantor:
University of Arizona

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleComputational Abstract Algebra: Using Monomial Matrices to Represent Groups in GAPen_US
dc.creatorRome, Zachary Roberten_US
dc.contributor.authorRome, Zachary Roberten_US
dc.date.issued2012-05-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractA monomial matrix is a matrix with exactly one non-zero element in each row and column. We will utilize GAP to construct all (transitive) representations of a given group using monomial matrices. First, essential group theory definitions and theorems will be provided, as well as an in-depth look at table of marks and monomial matrices. After describing the necessary mathematics, we will explore the GAP programming needed to achieve this goal. Ultimately we want a table, where each row represents a subgroup of the given group and, within the row, the table will hold the linear characters fixed by the monomial matrices of that subgroup. We will furthermore explore how to represent monomial matrices computationally in different ways and how to create a data structure to represent them. Our final goal will require GAP functions for finding all homomorphisms from a subgroup to roots of unity, using these homomorphisms to create monomial matrix representations of the group, and iterating through the subgroups of the group (up to conjugacy) to find all (transitive) monomial matrix representations of the group.en_US
dc.typetexten_US
dc.typeElectronic Thesisen_US
thesis.degree.nameB.S.en_US
thesis.degree.levelbachelorsen_US
thesis.degree.disciplineHonors Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.