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.
Keywords: metabelian groups; automorphic loops; Bruck loops; Moufang loops
DOI: DOI 10.14712/1213-7243.2020.043
AMS Subject Classification: 20N05