Automorphic loops and metabelian groups

TL;DR

This paper explores the properties of a specific commutative loop derived from uniquely 2-divisible groups, establishing conditions for it to be Moufang and providing partial evidence for a conjecture relating metabelian groups to automorphic loops.

Contribution

It characterizes when the loop is Moufang and confirms that split metabelian groups of odd order produce automorphic loops, advancing understanding of the conjecture.

Findings

The loop is Moufang under certain conditions.

Split metabelian groups of odd order yield automorphic loops.

Partial confirmation of Greer's conjecture.

Abstract

Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ)$ which arises as a result of a construction in \cite{baer}. We investigate some general properties and applications of $\circ$ and determine a necessary and sufficient condition on $G$ in order for $(G, \circ)$ to be Moufang. In \cite{greer14}, it is conjectured that $G$ is metabelian if and only if $(G, \circ)$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, \circ)$ is automorphic.