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  text: 'The kernel of a [-47t] $f: G \\to H$ is the collection of all elements $g$ in $G$ such that $f(g) = e_H$ the identity of $H$.\n\nIt is important to note that the kernel of any group homomorphism $G \\to H$ is always a subgroup of $G$.\nIndeed:\n\n- if $f(g_1) = e_H$ and $f(g_2) = e_H$ then $e_H = f(g_1) f(g_2) = f(g_1 g_2)$, so the kernel is closed under $G$'s operation;\n- if $f(x) = e_H$ then $e_H = f(e_G) = f(x^{-1} x) = f(x^{-1}) f(x) = f(x^{-1})$ (where we have used that [49z the image of the identity is the identity]), so inverses are contained in the putative subgroup;\n- $f(e_G) = e_H$ because the image of the identity is the identity, so the identity is contained in the putative subgroup.\n\nIt turns out that the notion of "[-4h6]" coincides exactly with the notion of "kernel of homomorphism". ([4h7 Proof.])\nThe "kernel of homomorphism" viewpoint of normal subgroups is much more strongly motivated from the point of view of [-4c7]; Timothy Gowers [considers this to be the correct way](https://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups/) to introduce the teaching of normal subgroups in the first place.',
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