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Definition (Identification Graph Using Cosets)
Let $G$ be a group, and let $\mathcal{G}=(G,S)$ be the Cayley graph of $G$ using the elements in $S$ to construct the colored arrows. Let $H$ be a subgroup of $G$. The identification graph of $G$ using the right cosets of $H$ is a colored directed graph such that
- The set of vertices is the set of right cosets $\{Ha\mid a\in G\}$, and
- We draw a colored arrow from the coset $Ha$ to the coset $Hb$ if and only if there exists a colored arrow $(c,d)$ with $c\in Ha$ and $d\in Hb$ in the original Cayley graph.
If instead of using right cosets, we used left cosets $aH$ as the vertices, then we'd call the graph the identification graph of $G$ using the left cosets of $H$.
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