# A related question to “relationship between generating elements given by cycles in Cayley graphs”

I am writing this question with reference to the post at Relationship between generating elements given by cycles in Cayley graph

When considering the generating elements $$g_qg_p$$, does it have the form $$(g_q,g_p) \in (\mathbb{Z}/(q−1)\mathbb{Z})\times (\mathbb{Z}/(p−1)\mathbb{Z})$$ ?

Suppose I consider the notion of the strong RSA assumption to some other group, such as $$(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q$$. Suppose the strong RSA assumption holds, at least with respect to finding the Hamiltonian cycle in Cayley graph which is well known to be a difficult problem.

Then for a generating elements set $$\{g_q,g_p\}$$, where $$g_p \in (\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \{0\}$$ and $$g_q \in (\{0\}) \rtimes \mathbb{Z}_q$$, I can do the computations in the same way by taking $$y=g_p^{y_p}g_q^{y_q}$$, $$z=g_p^{z_p}g_q^{z_q}$$, and $$g_p^{ry_p} \equiv g_p^{z_p} (mod p)$$, $$g_q^{ry_q} \equiv g_q^{z_q} (mod q)$$, right?

Am I correct? Have I understood it right?