{
localUrl: '../page/normal_subgroup.html',
arbitalUrl: 'https://arbital.com/p/normal_subgroup',
rawJsonUrl: '../raw/4h6.json',
likeableId: '2826',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
'EricBruylant'
],
pageId: 'normal_subgroup',
edit: '5',
editSummary: '',
prevEdit: '4',
currentEdit: '5',
wasPublished: 'true',
type: 'wiki',
title: 'Normal subgroup',
clickbait: 'Normal subgroups are subgroups which are in some sense "the same from all points of view".',
textLength: '747',
alias: 'normal_subgroup',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'PatrickStevens',
editCreatedAt: '2016-06-18 15:43:43',
pageCreatorId: 'PatrickStevens',
pageCreatedAt: '2016-06-17 10:27:44',
seeDomainId: '0',
editDomainId: 'AlexeiAndreev',
submitToDomainId: '0',
isAutosave: 'false',
isSnapshot: 'false',
isLiveEdit: 'true',
isMinorEdit: 'false',
indirectTeacher: 'false',
todoCount: '1',
isEditorComment: 'false',
isApprovedComment: 'true',
isResolved: 'false',
snapshotText: '',
anchorContext: '',
anchorText: '',
anchorOffset: '0',
mergedInto: '',
isDeleted: 'false',
viewCount: '34',
text: 'A *normal subgroup* $N$ of group $G$ is one which is closed under [4gk conjugation]: for all $h \\in G$, it is the case that $\\{ h n h^{-1} : n \\in N \\} = N$.\nIn shorter form, $hNh^{-1} = N$.\n\nSince [4gk conjugacy is equivalent to "changing the worldview"], a normal subgroup is one which "looks the same from the point of view of every element of $G$".\n\nA subgroup of $G$ is normal if and only if it is the [49y kernel] of some [-47t] from $G$ to some group $H$. ([4h7 Proof.])\n\n%%%knows-requisite([category_theory_equaliser]):\nFrom a category-theoretic point of view, the kernel of $f$ is an equaliser of an arrow $f$ with the zero arrow; it is therefore universal such that composition with $f$ yields zero.\n%%%\n\n[todo: why are they interesting]',
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: [
'subgroup_normal_iff_kernel_of_homomorphism',
'quotient_by_subgroup_is_well_defined_iff_normal',
'subgroup_normal_iff_union_of_conjugacy_classes'
],
parentIds: [
'group_mathematics'
],
commentIds: [],
questionIds: [],
tagIds: [
'definition_meta_tag',
'stub_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [
{
id: '4108',
parentId: 'group_conjugate',
childId: 'normal_subgroup',
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: '13934',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '5',
type: 'newEdit',
createdAt: '2016-06-18 15:43:43',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13717',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '4',
type: 'newChild',
createdAt: '2016-06-17 21:37:34',
auxPageId: 'subgroup_normal_iff_union_of_conjugacy_classes',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13718',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '4',
type: 'newRequiredBy',
createdAt: '2016-06-17 21:37:34',
auxPageId: 'subgroup_normal_iff_union_of_conjugacy_classes',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13630',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '4',
type: 'newRequiredBy',
createdAt: '2016-06-17 18:06:49',
auxPageId: 'simple_group',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13588',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '4',
type: 'newEdit',
createdAt: '2016-06-17 17:14:31',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13537',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '3',
type: 'newEdit',
createdAt: '2016-06-17 15:16:07',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13479',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '2',
type: 'newChild',
createdAt: '2016-06-17 10:58:44',
auxPageId: 'quotient_by_subgroup_is_well_defined_iff_normal',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13480',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '2',
type: 'newRequiredBy',
createdAt: '2016-06-17 10:58:44',
auxPageId: 'quotient_by_subgroup_is_well_defined_iff_normal',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13474',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '2',
type: 'newEdit',
createdAt: '2016-06-17 10:28:49',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13470',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '1',
type: 'newRequiredBy',
createdAt: '2016-06-17 10:27:59',
auxPageId: 'subgroup_normal_iff_kernel_of_homomorphism',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13469',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '1',
type: 'newEdit',
createdAt: '2016-06-17 10:27:44',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13463',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '1',
type: 'newRequirement',
createdAt: '2016-06-17 10:27:32',
auxPageId: 'group_conjugate',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13462',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '1',
type: 'newTag',
createdAt: '2016-06-17 10:27:18',
auxPageId: 'definition_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13461',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '1',
type: 'newTag',
createdAt: '2016-06-17 10:27:15',
auxPageId: 'stub_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13460',
pageId: 'normal_subgroup',
userId: 'PatrickStevens',
edit: '1',
type: 'newParent',
createdAt: '2016-06-17 10:27:07',
auxPageId: 'group_mathematics',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'true',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}