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text: 'Let $H$ be a subgroup of the [-3gd] $G$.\nThen $H$ is [4h6 normal] in $G$ if and only if it can be expressed as a [set_union union] of [4bj conjugacy classes].\n\n# Proof\n\n$H$ is normal in $G$ if and only if $gHg^{-1} = H$ for all $g \\in G$; equivalently, if and only if $ghg^{-1} \\in H$ for all $h \\in H$ and $g \\in G$.\n\nBut if we fix $h \\in H$, then the statement that $ghg^{-1} \\in H$ for all $g \\in G$ is equivalent to insisting that the conjugacy class of $h$ is contained in $H$.\nTherefore $H$ is normal in $G$ if and only if, for all $h \\in H$, the conjugacy class of $h$ lies in $H$.\n\nIf $H$ is normal, then it is clearly a union of conjugacy classes (namely $\\cup_{h \\in H} C_h$, where $C_h$ is the conjugacy class of $h$).\n\nConversely, if $H$ is not normal, then there is some $h \\in H$ such that the conjugacy class of $h$ is not wholly in $H$; so $H$ is not a union of conjugacy classes because it contains $h$ but not the entire conjugacy class of $h$.\n(Here we have used that the [-conjugacy_classes_partition_the_group].)\n\n# Interpretation\n\nA normal subgroup is one which is fixed under conjugation; the most natural (and, indeed, the smallest) objects which are fixed under conjugation are conjugacy classes; so this criterion tells us that to obtain a *subgroup* which is fixed under conjugation, it is necessary and sufficient to assemble these objects (the conjugacy classes), which are themselves the smallest objects which are fixed under conjugation, into a group.',
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