Date of Award

6-1-2015

Document Type

Dissertation

Degree Name

Ph.D.

Organizational Unit

College of Natual Science and Mathematics

First Advisor

Petr Vojtěchovský, Ph.D.

Second Advisor

Scott Leutenegger

Third Advisor

Nikolaos Galatos

Fourth Advisor

Michael Kinyon

Fifth Advisor

Andrew Linshaw

Sixth Advisor

Izabella Stuhl

Keywords

Automorphic loops, Dihedral-like automorphic loops, Cauchy and Lagrange theorems for groups

Abstract

In this dissertation we study dihedral-like constructions of automorphic loops. Automorphic loops are loops in which all inner mappings are automorphisms. We start by describing a generalization of the dihedral construction for groups. Namely, if (G , +) is an abelian group, m > 1 and α ∈2 Aut(G ), let Dih(m, G, α) on Zm × G be defined by

(i, u )(j, v ) = (i + j , ((-1)j u + v )αij ).

We prove that the resulting loop is automorphic if and only if m = 2 or (α2 = 1 and m is even) or (m is odd, α = 1 and exp(G ) ≤ 2). In the last case, the loop is a group. The case m = 2 was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský.

We study basic structural properties of dihedral-like automorphic loops. We describe certain subloops, including: nucleus, commutant, center, associator subloop and derived subloop. We prove theorems for dihedral-like automorphic loops analogous to the Cauchy and Lagrange theorems for groups, and further we discuss the coset decomposition in dihedral-like automorphic loops.

We show that two finite dihedral-like automorphic loops Dih( m, G, α) and Dih(m̅, Ḡ, ᾱ) are isomorphic if and only if m= m̅, G [congruent with] Ḡ and α is conjugate to ᾱ in Aut(G ). We describe the automorphism group of Q and its subgroup consisting of inner mappings of Q .

Finally, due to the solution to the isomorphism problem, we are interested in studying conjugacy classes of automorphism groups of finite abelian groups. Then we describe all dihedral-like automorphic loops of order < 128 up to isomorphism. We conclude with a description of all dihedral-like automorphic loops of order < 64 up to isotopism.

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Mouna Ramadan Aboras

Provenance

Received from ProQuest

File Format

application/pdf

Language

en

File Size

101 p.

Discipline

Mathematics

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