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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be [[subgroups|0X1JXT47GA]] of $G$.\nThe product of $H$ and $K$ is the subset $HK = \\{hk : h \\in H, k \\in K\\}$ of $G$.\n\n## Remarks\n\nThe set $HK$ is a subgroup of $G$ if and only if $HK = KH$.\nIt need not be a subgroup. In the symmetric group $S_3$, two distinct subgroups of order $2$ have a product with $4$ elements, and $4$ does not divide $6$.\nWhen $HK$ is a subgroup, it is the [[subgroup generated by|0RP7VWQ0YN]] $H \\cup K$.\n\n## Explanations\n\n### L1 human\n\nTo multiply two subgroups, multiply each element of the first by each element of the second.\nThe result contains both subgroups, but it may fail to be a subgroup, because a product $hk h'k'$ need not have the form $h''k''$.\nIn an abelian group, the product of two subgroups is always a subgroup.\n",
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  "name": "Product of two subgroups",
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