Let $S$ generate a finite group $G$ and $s \in S$ such that $\langle s\rangle \trianglelefteq G$, ${\rm Cay}(G/\langle s\rangle,S)$ has a Hamiltonian cycle.
Let $(s_1,s_2, \cdots, s_n)$ be the Hamiltonian cycle in ${\rm Cay}(G/\langle s\rangle,S)$ and $k=|s_1 s_2 \cdots s_n|$.
Let $m=|G|/(nk)$
Claim: $(s^{m-1}, s_1, s^{m-1}, s_2, s^{m-1}, \cdots, s^{m-1}, s_n)^k$ is a Hamiltonian cycle in ${\rm Cay}(G,S)$.
In the above, how can I understand regarding the hamiltonian cycle expressed by the claim and the value $m$?
Like, can I think like, if $|G|$ can be a value like $pq$, and $|s|=q$, and then $|k|=q$, and $m=pq/q q$? What type of values will be occupied by $m$ and will it be traversing the vertices belonging to cosets of $\langle s\rangle$, when we take $s^{m-1}$?
Thanks a lot in advance.