Notes on the hidden subgroup problem on some semi-direct product groups

We consider the hidden subgroup problem on the semi-direct product of cyclic groups $\Z_{N}\rtimes\Z_{p}$ with some restriction on $N$ and $p$. By using the homomorphic properties, we present a class of semi-direct product groups in which the structures of subgroups can be easily classified. Furthermore, we show that there exists an efficient quantum algorithm for the hidden subgroup problem on the class.