{ localUrl: '../page/kernel_of_group_homomorphism.html', arbitalUrl: 'https://arbital.com/p/kernel_of_group_homomorphism', rawJsonUrl: '../raw/49y.json', likeableId: '2680', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'kernel_of_group_homomorphism', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'Kernel of group homomorphism', clickbait: '', textLength: '1141', alias: 'kernel_of_group_homomorphism', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-17 15:06:36', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-14 19:36:09', 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: '46', 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.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'true', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'PatrickStevens' ], childIds: [], parentIds: [ 'group_homomorphism' ], commentIds: [], questionIds: [], tagIds: [ 'needs_clickbait_meta_tag', 'definition_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [ { id: '3891', parentId: 'group_homomorphism', childId: 'kernel_of_group_homomorphism', type: 'requirement', creatorId: 'AlexeiAndreev', createdAt: '2016-06-17 21:58:56', level: '1', isStrong: 'false', everPublished: 'true' } ], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17121', pageId: 'kernel_of_group_homomorphism', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-19 02:03:23', auxPageId: 'needs_clickbait_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13530', pageId: 'kernel_of_group_homomorphism', userId: 'PatrickStevens', edit: '2', type: 'newEdit', createdAt: '2016-06-17 15:06:36', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13472', pageId: 'kernel_of_group_homomorphism', userId: 'PatrickStevens', edit: '1', type: 'newRequiredBy', createdAt: '2016-06-17 10:28:02', auxPageId: 'subgroup_normal_iff_kernel_of_homomorphism', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12770', pageId: 'kernel_of_group_homomorphism', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-06-14 19:36:09', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12736', pageId: 'kernel_of_group_homomorphism', userId: 'PatrickStevens', edit: '1', type: 'newTag', createdAt: '2016-06-14 19:11:31', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12735', pageId: 'kernel_of_group_homomorphism', userId: 'PatrickStevens', edit: '1', type: 'newRequirement', createdAt: '2016-06-14 19:11:25', auxPageId: 'group_homomorphism', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12734', pageId: 'kernel_of_group_homomorphism', userId: 'PatrickStevens', edit: '1', type: 'newParent', createdAt: '2016-06-14 19:11:19', auxPageId: 'group_homomorphism', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }