On the number of transversals in a class of Latin squares

Denote by Apk the Latin square of order n=pk formed by the Cayley table of the additive group (Zpk,+), where p is an odd prime and k is a positive integer. It is shown that for each p there exists Q>0 such that for all sufficiently large k, the number of transversals in Apk exceeds (nQ)np(p−1).