#### Date of Award

1-1-2015

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Petr Vojtěchovský

#### 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 Z*m* × *G* be defined by

(*i, u* )(*j, v* ) = (*i + j* , ((-1)*j** u* + *v* )α* ij* ).

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 Vojtechovský.

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(*m ̄, G ̄,* [special character omitted]) are isomorphic if and only if *m* = *m ̄* , *G* [congruent with] *G ̄* and α is conjugate to [special character omitted] 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.

#### Recommended Citation

Aboras, Mouna Ramadan, "Dihedral-like constructions of automorphic loops" (2015). *Electronic Theses and Dissertations*. 3.

http://digitalcommons.du.edu/etd/3

#### Provenance

Recieved from ProQuest

#### Rights holder

Mouna Ramadan Aboras

#### File size

101 p.

#### Copyright date

2015

#### File format

application/pdf

#### Language

en

#### Discipline

Mathematics