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Let $\pi: G\to G_1$ be a surjective group homomorphism to a finite group $G_1$ and let $S_1$ be a (finite) generating set of $G_1$. Assume $\mathrm{Cay}(G_1,S_1)$ is the Cayley graph of $G_1$ with respect to the generating set $S_1$. Define $S:=\pi^{-1}(S_1)$ and consider the Cayley graph $\mathrm{Cay}(G,S)$.

I would like to know if there is any criteria that allows one to construct an embedding of $\mathrm{Cay}(G_1,S_1)$ into $\mathrm{Cay}(G,S)$. Surly this is not always possible but I am looking for an example that existence of such embedding is possible. For example when $\pi: \mathrm{GL}_2(\mathbb{Z}_p)\to \mathrm{GL}_2(\mathbb{F}_p)$ is the reduction map and one can use for instance Hensel lemma to construct fibers.

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This is possible (and easy), because you have defined $S$ to be the entire inverse image of $S_1$. (An embedding is unlikely to exist if $S$ is some small subset of $G$ that is mapped onto $S_1$ by $\pi$.)

For each $g_1 \in G_1$, let $\varphi(g_1)$ be any element of $\pi^{-1}(g_1)$. Then $\varphi$ is an embedding of $\mathrm{Cay}(G_1,S_1)$ onto an induced subgraph of $\mathrm{Cay}(G,S)$: for $g_1, h_1 \in G_1$, we have $$ \varphi(g_1) - \varphi(h_1) \Leftrightarrow \varphi(g_1) \varphi(h_1)^{-1} \in S = \pi^{-1}(S_1) \Leftrightarrow \pi \bigl( \varphi(g_1) \varphi(h_1)^{-1} \bigr) \in S_1 \Leftrightarrow g_1 h_1^{-1} \in S_1 \Leftrightarrow g_1 - h_1 .$$

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  • $\begingroup$ Another way to say what Dave said is that, since $S$ is the full preimage of $S_1$, as a graph, $\mathrm{Cay}(G,S)$ is the lexicographic product of $\mathrm{Cay}(G_1,S_1)$ with the edgeless graph on $G/G_1$ (and thus any choice of one vertex per preimage will induce an embedding). $\endgroup$
    – verret
    Commented Mar 11, 2015 at 8:21

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