# Question regarding the factor group lemma for Cayley graphs

Can someone please explain the proof of the "Factor group lemma" for Cayley graphs which is stated below.

Factor Group Lemma: Suppose that

1.$$N$$ is a cyclic, normal subgroup of a group $$G$$.

2.$$(s_1,s_2,\ldots,s_m)$$ is a hamiltonian cycle in $$Cay(G/N;S)$$.

3.The product $$s_1s_2\cdots s_m$$ generates $$N$$.

Then $$Cay(G;S)$$ has a Hamiltonian cycle.

• The formulation is somewhat imprecise (or lacking rigour). It would appear that $S$ is at the same time a subset of both the support set $G$ as well as the quotient-set $G/N$. Perhaps you meant to refer to the image of $S$ under the canonical surjection into the quotient group? The same issue affects the vertices $s_k$.