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Definition (Cayley Graph Of A Span Of Permutations)
Let $X$ be a set. Let $S$ be a set of permutations of $X$. Then a Cayley graph of $H=\text{span} (S)$ generated by $S$, which we'll write as $(H,S)$, is a colored directed graph that satisfies the the following three properties:
- The vertex set is $\text{span} (S)$. Each vertex corresponds to a permutation.
- Each element $s \in S$ is assigned a unique color which we'll denote by $c_s$.
- For each color $c_s$, and each vertex $\sigma$, we draw the colored arrow $(\sigma ,s \circ \sigma)$.
Most of the time we assume that $S$ does not contain the identity. However, if it does contain the identity, then we just draw a colored loop $(\sigma,\sigma)$ at each vertex.
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