Finite loops with dihedral inner mapping groups are solvable. (Q1427390)

Let \(Q\) be a loop, let \(M(Q)\) be its multiplication group, and \(I(Q)\) be its inner mapping group. The author proves the following new results. If \(I(Q)\) is a finite dihedral group then \(M(Q)\) is a solvable group, and in the finite case \(Q\) is a solvable loop. Assume that \(G\) is a finite group, \(H\) a dihedral subgroup of \(G\), and assume that there exist \(H\)-connected transversals in \(G\). Then \(G\) is a solvable group.