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  "body": "\n## Example\n\nIn the [[symmetric group|0515A6D2Y1]] $S_3$, let $H = \\{e, (1\\,2)\\}$. Compose permutations from right to left.\nTwo of the [[left cosets|05FD317V2W]] are $(1\\,3)H = \\{(1\\,3), (1\\,2\\,3)\\}$ and $(2\\,3)H = \\{(2\\,3), (1\\,3\\,2)\\}$.\nTry to set $(aH)(bH) = (ab)H$. With $a = (1\\,3)$ and $b = (2\\,3)$, we get $ab = (1\\,3\\,2)$, which is in $(2\\,3)H$.\nWith $a = (1\\,2\\,3)$ from the same coset, we get $ab = (1\\,2)$, which is in $H$.\nThe two results are in different cosets. Thus this product is not defined, and the cosets of $H$ do not form a [[quotient group|0YYWCNTPWR]].\n\n## Remarks\n\n$H$ is not [[normal|0W77CVBCHQ]] in $S_3$. For a normal subgroup the product of cosets is always defined.\n",
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  "name": "The cosets of a subgroup that is not normal do not form a group",
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