## Electronic Theses and Dissertations

6-1-2015

Dissertation

Ph.D.

#### Department

Mathematics

Petr Vojtěchovský, Ph.D.

#### 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 + vij ).

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] and α is conjugate to 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.

101 p.

application/pdf

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Mathematics

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