On right conjugacy closed loops and right conjugacy closed loop folders

• Über rechts-konjugationsinvariante Loops und rechts-konjugationsinvariante Loop-Folder

Artic, Katharina; Hiß, Gerhard (Thesis advisor); Niemeyer, Alice C. (Thesis advisor)

Aachen (2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

Loops allow us to study structures with a possibly non-associative multiplication. They have been researched intensely since the beginning of last century. There is a known relationship between loops and groups via group theoretic objects named loop folders. This relationship allows us to examine loops with group theoretic methods. A right conjugacy closed loop is a loop whose set of right multiplications is closed under conjugation. We call such loops RCC loops.This thesis examines RCC loops and RCC loop folders. After presenting some basic properties of RCC loop folders we use the relationship between loops and groups to examine RCC loops by considering their envelopes. We show that the group PSL(2,q) is never isomorphic to the group of an RCC loop folder and give a new group theoretic proof of a known result of Aleš Drápal: An RCC loop of prime order is associative. The idea of this proof is generalized to show that the right multiplication group of a non-associative RCC loop of order p1p2 where p1 and p2 are distinct primes, is an imprimitive group. Further we present an algorithm to compute the envelopes of non-associativeRCC loops. This algorithm has been used to compute all RCC loops of an order up to 30. A database of these RCC loops is compiled and available in the open source computer algebra system GAP via the package LOOPS} by Gábor P. Nagy and Petr Vojtěchovsky.