Abstract:Let $G$ be a finite group with a dicyclic subgroup $H$. We show that if there exist $H$-connected transversals in $G$, then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I(Q)$ of a~finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I(Q)$ is a certain type of a direct product.
Keywords: solvable loop; inner mapping group; dicyclic group
DOI: DOI 10.14712/1213-7243.2015.180
AMS Subject Classification: 20N05 20D10