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 ``Engelsche elemente noetherscher gruppen'' (1957) by R. 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 ``A class of loops categorically isomorphic to Bruck loops of odd order'' (2014) by M. Greer, 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.