Automorphic loops and metabelian groups

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.