On Connected Transversals to Nonabelian Subgroups

In this paper we consider groups G which have connected transversals to nonabelian subgroups whose order is a product of two odd primes p and q, where p > q andp = 2 qm + 1. In our main theorem we show thatG is then solvable. We apply our results to loop theory and it follows that if the inner mapping group of a finite loop has order pq, where p and q are as previously given, then the loop is solvable.