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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.\nThe quotient group $G/N$ is the set of [[left cosets|03DEH43F4H]] $\\{gN : g \\in G\\}$ with the operation $(aN)(bN) = (ab)N$.\n\n## Remarks\n\nThe operation is well defined because $N$ is normal: if $aN = a'N$ and $bN = b'N$, then $abN = a'b'N$.\nThe identity of $G/N$ is $N = eN$, and the inverse of $aN$ is $a^{-1}N$.\nWhen $G$ is finite, $|G/N| = [G : N] = |G| / |N|$.\nThe map $g \\mapsto gN$ is a surjective homomorphism from $G$ to $G/N$ with kernel $N$.\n\n## Explanations\n\n### L1 human\n\nA quotient group treats all elements of one coset as one element.\nFor example, the integers modulo $2$ come from the integers when you treat all even numbers as one element and all odd numbers as another.\nThe normality of $N$ makes sure that the product of two cosets does not depend on the chosen elements.\n",
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