{ localUrl: '../page/subgroup_normal_iff_union_of_conjugacy_classes.html', arbitalUrl: 'https://arbital.com/p/subgroup_normal_iff_union_of_conjugacy_classes', rawJsonUrl: '../raw/4jw.json', likeableId: '2756', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'subgroup_normal_iff_union_of_conjugacy_classes', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Subgroup is normal if and only if it is a union of conjugacy classes', clickbait: 'A useful way to think about normal subgroups, which meshes with their "closed under conjugation" interpretation.', textLength: '1483', alias: 'subgroup_normal_iff_union_of_conjugacy_classes', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-17 21:37:34', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-17 21:37:34', 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: '25', 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.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', 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: [ 'normal_subgroup' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [ { id: '4199', parentId: 'normal_subgroup', childId: 'subgroup_normal_iff_union_of_conjugacy_classes', 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: '13729', pageId: 'subgroup_normal_iff_union_of_conjugacy_classes', userId: 'PatrickStevens', edit: '1', type: 'newRequiredBy', createdAt: '2016-06-17 21:39:52', auxPageId: 'alternating_group_five_is_simple', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13722', pageId: 'subgroup_normal_iff_union_of_conjugacy_classes', userId: 'PatrickStevens', edit: '1', type: 'newRequiredBy', createdAt: '2016-06-17 21:38:43', auxPageId: 'simplicity_of_alternating_group_five_simpler_proof', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13719', pageId: 'subgroup_normal_iff_union_of_conjugacy_classes', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-06-17 21:37:34', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13716', pageId: 'subgroup_normal_iff_union_of_conjugacy_classes', userId: 'PatrickStevens', edit: '1', type: 'newRequirement', createdAt: '2016-06-17 21:26:59', auxPageId: 'normal_subgroup', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13715', pageId: 'subgroup_normal_iff_union_of_conjugacy_classes', userId: 'PatrickStevens', edit: '1', type: 'newParent', createdAt: '2016-06-17 21:26:55', auxPageId: 'normal_subgroup', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }