{ localUrl: '../page/subgroup_normal_iff_kernel_of_homomorphism.html', arbitalUrl: 'https://arbital.com/p/subgroup_normal_iff_kernel_of_homomorphism', rawJsonUrl: '../raw/4h7.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'subgroup_normal_iff_kernel_of_homomorphism', edit: '3', editSummary: '', prevEdit: '2', currentEdit: '3', wasPublished: 'true', type: 'wiki', title: 'Subgroup is normal if and only if it is the kernel of a homomorphism', clickbait: 'The "correct way" to think about normal subgroups is as kernels of homomorphisms.', textLength: '1446', alias: 'subgroup_normal_iff_kernel_of_homomorphism', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-17 10:33:06', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-17 10:26:06', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '39', text: 'Let $N$ be a [-subgroup] of [-3gd] $G$.\nThen $N$ is [4h6 normal] in $G$ if and only if there is a group $H$ and a [-47t] $\\phi:G \\to H$ such that the [49y kernel] of $\\phi$ is $N$.\n\n# Proof\n\n## "Normal" implies "is a kernel"\nLet $N$ be normal, so it is closed under [4gk conjugation].\nThen we may define the [-quotient_group] $G/N$, whose elements are the [group_coset left cosets] of $N$ in $G$, and where the operation is that $gN + hN = (g+h)N$.\nThis group is well-defined ([quotient_by_subgroup_is_well_defined_iff_normal proof]).\n\nNow there is a homomorphism $\\phi: G \\to G/N$ given by $g \\mapsto gN$.\nThis is indeed a homomorphism, by definition of the group operation $gN + hN = (g+h)N$.\n\nThe kernel of this homomorphism is precisely $\\{ g : gN = N \\}$; that is simply $N$:\n\n- Certainly $N \\subseteq \\{ g : gN = N \\}$ (because $nN = N$ for all $n$, since $N$ is closed as a subgroup of $G$);\n- We have $\\{ g : gN = N \\} \\subseteq N$ because if $gN = N$ then in particular $g e \\in N$ (where $e$ is the group identity) so $g \\in N$.\n\n## "Is a kernel" implies "normal"\nLet $\\phi: G \\to H$ have kernel $N$, so $\\phi(n) = e$ if and only if $n \\in N$.\nWe claim that $N$ is closed under conjugation by members of $G$.\n\nIndeed, $\\phi(h n h^{-1}) = \\phi(h) \\phi(n) \\phi(h^{-1}) = \\phi(h) \\phi(h^{-1})$ since $\\phi(n) = e$.\nBut that is $\\phi(h h^{-1}) = \\phi(e)$, so $hnh^{-1} \\in N$.\n\nThat is, if $n \\in N$ then $hnh^{-1} \\in N$, so $N$ is normal.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', 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