On Connected Transversals to Subgroups Whose Order is a Product of Two Primes

We consider finite groups which have connected transversals to subgroups whose order is a product of two primespandq. We investigate those values ofpandqfor which the group is soluble. We can show that the solubility of the group follows ifq = 2 andp ≤ 61,q = 3 andp ≤ 31,q = 5 andp ≤ 11. We then apply our results on loop theory and we show that if the inner mapping group of a finite loop has orderpqwherepandqare as above then the loop is soluble.