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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $S$ be a [[generating set|0MBV2VHMXA]] of $G$.\nThe Cayley graph $\\Gamma(G, S)$ is the directed graph with vertex set $G$.\nFor each $g \\in G$ and $s \\in S$, it has an edge with colour $s$ from $g$ to $g s$.\n\n## Remarks\n\nFor $G = \\mathbb{Z}$ and $S = \\{1, -1\\}$, the Cayley graph is an infinite path. For $\\mathbb{Z}/n\\mathbb{Z}$ with a generator and its inverse, it is the cycle $C_n$.\nThe Cayley graph of a free group with respect to a basis is a tree.\nThe [[left multiplication action|0K5TWJAGPD]] of $G$ on itself acts on $\\Gamma(G, S)$, so each Cayley graph is vertex-transitive.\nArthur Cayley first considered these graphs for finite groups in 1878.\n\n## Explanations\n\n### L1 human\n\nA Cayley graph draws a group as a network.\nEach element is a point, and each generator gives an arrow from $g$ to $g s$.\nPaths in the graph correspond to words in the generators.\n",
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