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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \\in G$.\nThe left coset of $H$ with representative $g$ is the set $gH = \\{gh : h \\in H\\}$.\n\n## Remarks\n\nTwo left cosets $aH$ and $bH$ are equal when $a^{-1}b \\in H$, and they are disjoint otherwise.\nThus the left cosets of $H$ form a partition of $G$, and $H = eH$ is one of them.\nIn $(\\mathbb{Z}, +)$, the left cosets of $3\\mathbb{Z}$ are $3\\mathbb{Z}$, $1 + 3\\mathbb{Z}$ and $2 + 3\\mathbb{Z}$.\n\n## Explanations\n\n### L1 human\n\nA left coset is a copy of the subgroup that you move by multiplying with a fixed element on the left.\nDifferent elements can give the same coset.\nFor the even integers inside the integers, there are two cosets: the even integers and the odd integers.\n",
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