solvable loop; inner mapping group; dicyclic group
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.
 Leppälä E., Niemenmaa M.: On finite loops whose inner mapping groups are direct products of dihedral groups and abelian groups
. Quasigroups Related Systems 20 (2012), no. 2, 257–260. MR 3232747
| Zbl 1273.20073
 Leppälä E., Niemenmaa M.: On finite commutative loops which are centrally nilpotent
. Comment. Math. Univ. Carolin. 56 (2015), no. 2, 139–143. MR 3338728
| Zbl 1339.20064