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text: 'Let $G$ be a [-3gd] and $N$ a [-4h6] of $G$.\nThen we may define the *quotient group* $G/N$ to be the set of [4j4 left cosets] $gN$ of $N$ in $G$, with the group operation that $gN + hN = (gh)N$.\nThis is well-defined if and only if $N$ is normal.\n\n# Proof\n\n## $N$ normal implies $G/N$ well-defined\n\nRecall that $G/N$ is well-defined if "it doesn't matter which way we represent a coset": whichever coset representatives we use, we get the same answer.\n\nSuppose $N$ is a normal subgroup of $G$.\nWe need to show that given two representatives $g_1 N = g_2 N$ of a coset, and given representatives $h_1 N = h_2 N$ of another coset, that $(g_1 h_1) N = (g_2 h_2)N$.\n\nSo given an element of $g_1 h_1 N$, we need to show it is in $g_2 h_2 N$, and vice versa.\n\nLet $g_1 h_1 n \\in g_1 h_1 N$; we need to show that $h_2^{-1} g_2^{-1} g_1 h_1 n \\in N$, or equivalently that $h_2^{-1} g_2^{-1} g_1 h_1 \\in N$.\n\nBut $g_2^{-1} g_1 \\in N$ because $g_1 N = g_2 N$; let $g_2^{-1} g_1 = m$.\nSimilarly $h_2^{-1} h_1 \\in N$ because $h_1 N = h_2 N$; let $h_2^{-1} h_1 = p$.\n\nThen we need to show that $h_2^{-1} m h_1 \\in N$, or equivalently that $p h_1^{-1} m h_1 \\in N$.\n\nSince $N$ is closed under conjugation and $m \\in N$, we must have that $h_1^{-1} m h_1 \\in N$;\nand since $p \\in N$ and $N$ is closed under multiplication, we must have $p h_1^{-1} m h_1 \\in N$ as required.\n\n## $G/N$ well-defined implies $N$ normal\n\nFix $h \\in G$, and consider $hnh^{-1} N + hN$.\nSince the quotient is well-defined, this is $(hnh^{-1}h) N$, which is $hnN$ or $hN$ (since $nN = N$, because $N$ is a subgroup of $G$ and hence is closed under the group operation).\nBut that means $hnh^{-1}N$ is the identity element of the quotient group, since when we added it to $hN$ we obtained $hN$ itself.\n\nThat is, $hnh^{-1}N = N$.\nTherefore $hnh^{-1} \\in N$.\n\nSince this reasoning works for any $h \\in G$, it follows that $N$ is closed under conjugation by elements of $G$, and hence is normal.',
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