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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?

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