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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $S$ be a subset of $G$.\nThe *Schreier coset graph* of $(G, H, S)$ is the [[directed graph|0KFYGSHHF5]] with labelled edges such that:\n\n1. its vertices are the [[right cosets|0NE1DK7H4T]] $Hg$ of $H$ in $G$;\n2. for each vertex $Hg$ and each $s \\in S$, there is one edge with label $s$ from $Hg$ to $Hgs$.\n\n## Remarks\n\nThe rule is well defined: if $Hg = Hg'$, then $g' = hg$ for some $h \\in H$, so $Hg's = Hhgs = Hgs$.\nThe number of vertices is the index of $H$ in $G$, and every vertex has exactly one outgoing edge with each label.\nIf $S$ generates $G$, the graph is connected.\nFor $H = \\set{1}$ it is the [[Cayley graph|0GYQK9NDJF]] of $G$ with respect to $S$.\n\nExample: let $G = \\Z$, $H = 3\\Z$ and $S = \\set{1}$.\nThe vertices are $3\\Z$, $1 + 3\\Z$ and $2 + 3\\Z$, and the edges go $3\\Z \\to 1 + 3\\Z \\to 2 + 3\\Z \\to 3\\Z$, since adding $1$ moves each coset to the next one.\nThe graph is a directed cycle of length $3$.\n\nNon-example: the rule with left cosets $gH$ and edges $gH \\to gsH$ is not well defined in general.\nIn $S_3$, with composition from right to left, take $H = \\set{1, (1\\,2)}$ and $s = (2\\,3)$.\nThen $1 \\cdot H = (1\\,2) H$, but $(2\\,3) H = \\set{(2\\,3), (1\\,3\\,2)}$ and $(1\\,2)(2\\,3) H = (1\\,2\\,3) H = \\set{(1\\,2\\,3), (1\\,3)}$ are different cosets.\n",
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