On connected transversals to dihedral subgroups (Q2713282)

Let \(G\) be a group with a dihedral subgroup \(H\) of order \(2x\), where \(x\) is an odd number. It is shown that, if there exist \(H\)-connected transversals \(A\) and \(B\) in \(G\), then \(G\) is a solvable group. (A subset \(A\) of \(G\) is said to be a left transversal to \(H\) if it contains exactly one element of each left coset of \(H\). If moreover \([A,B]<H\), then the transversals \(A\) and \(B\) are said to be \(H\)-connected.) This result applies to loop theory. It is proved that, if the inner mapping group \(I(Q)\) of a finite loop \(Q\) is dihedral of order \(2x\), then \(Q\) is a solvable loop.