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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $N$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThen $N$ is [[normal|0CB0DX5BVV]] in $G$ if and only if $gN = Ng$ for each $g \\in G$.\n\n## Remarks\n\nHere $gN$ is a [[left coset|03DEH43F4H]] and $Ng$ is a [[right coset|0NE1DK7H4T]].\nThe equation $gN = Ng$ is equivalent to $gNg^{-1} = N$, by multiplying on the right by $g^{-1}$.\nThe condition does not say that $gn = ng$ for each $n \\in N$; it says only that the two sets are equal.\n\n## Explanations\n\n### L1 human\n\nFor a normal subgroup, it does not matter if you multiply by $g$ on the left or on the right: you get the same coset.\nThus there is only one kind of coset, and we can just say \"the cosets of $N$\".\nThis is what makes the quotient group possible.\n",
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  "name": "Normality via left and right cosets",
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