{
localUrl: '../page/abelian_group.html',
arbitalUrl: 'https://arbital.com/p/abelian_group',
rawJsonUrl: '../raw/3h2.json',
likeableId: '2503',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
'EricRogstad'
],
pageId: 'abelian_group',
edit: '16',
editSummary: 'Fixing commutative_operation greenlinks',
prevEdit: '15',
currentEdit: '16',
wasPublished: 'true',
type: 'wiki',
title: 'Abelian group',
clickbait: 'A group where the operation commutes. Named after Niels Henrik Abel. ',
textLength: '2370',
alias: 'abelian_group',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'EricRogstad',
editCreatedAt: '2016-07-18 17:59:32',
pageCreatorId: 'NateSoares',
pageCreatedAt: '2016-05-09 06:11:40',
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: '94',
text: '[summary: An abelian group is a [3gd group] where the operation is [3jb commutative]. That is, an abelian group $G$ is a pair $(X, \\bullet)$ where $X$ is a [3jz set] and $\\bullet$ is a binary [3h7 operation] obeying the four group axioms plus an axiom of commutativity:\n\n1. [3gy Closure]: For all $x, y$ in $X$, $x \\bullet y$ is defined and in $X$. We abbreviate $x \\bullet y$ as $xy$.\n2. [3h4 Associativity]: $x(yz) = (xy)z$ for all $x, y, z$ in $X$.\n3. Identity: There is an element $e$ such that for all $x$ in $X$, $xe=ex=x$.\n4. Inverses: For each $x$ in $X$ is an element $x^{-1}$ in $X$ such that $xx^{-1}=x^{-1}x=e$.\n5. [3jb Commutativity]: For all $x, y$ in $X$, $xy=yx$.\n \nAbelian groups are very "well-behaved" groups that are often easier to deal with than their non-commuting counterparts.]\n\nAn abelian group is a [3gd group] $G=(X, \\bullet)$ where $\\bullet$ is [3jb commutative]. In other words, the group operation satisfies the five axioms:\n\n1. [3gy Closure]: For all $x, y$ in $X$, $x \\bullet y$ is defined and in $X$. We abbreviate $x \\bullet y$ as $xy$.\n2. [3h4 Associativity]: $x(yz) = (xy)z$ for all $x, y, z$ in $X$.\n3. Identity: There is an element $e$ such that for all $x$ in $X$, $xe=ex=x$.\n4. Inverses: For each $x$ in $X$ is an element $x^{-1}$ in $X$ such that $xx^{-1}=x^{-1}x=e$.\n5. [3jb Commutativity]: For all $x, y$ in $X$, $xy=yx$.\n\nThe first four are the standard [3gd group axioms]; the fifth is what distinguishes abelian groups from groups. \n\nCommutativity gives us license to re-arrange chains of elements in formulas about commutative groups. For example, if in a commutative group with elements $\\{1, a, a^{-1}, b, b^{-1}, c, c^{-1}, d\\}$, we have the claim $aba^{-1}db^{-1}=d^{-1}$, we can shuffle the elements to get $aa^{-1}bb^{-1}d=d^{-1}$ and reduce this to the claim $d=d^{-1}$. This would be invalid for a nonabelian group, because $aba^{-1}$ doesn't necessarily equal $aa^{-1}b$ in general.\n\nAbelian groups are very well-behaved groups, and they are often much easier to deal with than their non-commutative counterparts. For example, every [576] of an abelian group is [4h6 normal], and all finitely generated abelian groups are a [group_theory_direct_product direct product] of [47y cyclic groups] (the [structure_theorem_for_finitely_generated_abelian_groups structure theorem for finitely generated abelian groups]). ',
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: [
'EricRogstad',
'NateSoares',
'QiaochuYuan',
'AlexeiAndreev'
],
childIds: [],
parentIds: [
'group_mathematics',
'algebraic_structure'
],
commentIds: [
'3sy'
],
questionIds: [],
tagIds: [],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [],
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: '17069',
pageId: 'abelian_group',
userId: 'EricRogstad',
edit: '16',
type: 'newEdit',
createdAt: '2016-07-18 17:59:32',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: 'Fixing commutative_operation greenlinks'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '16262',
pageId: 'abelian_group',
userId: 'EricRogstad',
edit: '15',
type: 'newEdit',
createdAt: '2016-07-08 21:54:58',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: 'fixed Associativity greenlink'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14985',
pageId: 'abelian_group',
userId: 'EricRogstad',
edit: '14',
type: 'newEdit',
createdAt: '2016-06-30 18:06:07',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: 'Harmonize summary and first sentence'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13066',
pageId: 'abelian_group',
userId: 'PatrickStevens',
edit: '13',
type: 'newRequiredBy',
createdAt: '2016-06-15 14:59:08',
auxPageId: 'dihedral_groups_are_non_abelian',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10902',
pageId: 'abelian_group',
userId: 'QiaochuYuan',
edit: '13',
type: 'newEdit',
createdAt: '2016-05-25 20:34:09',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10899',
pageId: 'abelian_group',
userId: 'QiaochuYuan',
edit: '0',
type: 'newAlias',
createdAt: '2016-05-25 20:30:26',
auxPageId: '',
oldSettingsValue: 'commutative_group',
newSettingsValue: 'abelian_group'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10900',
pageId: 'abelian_group',
userId: 'QiaochuYuan',
edit: '12',
type: 'newEdit',
createdAt: '2016-05-25 20:30:26',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10359',
pageId: 'abelian_group',
userId: 'EricRogstad',
edit: '11',
type: 'newEdit',
createdAt: '2016-05-14 20:33:37',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10358',
pageId: 'abelian_group',
userId: 'EricRogstad',
edit: '10',
type: 'newEdit',
createdAt: '2016-05-14 20:31:11',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10357',
pageId: 'abelian_group',
userId: 'EricRogstad',
edit: '9',
type: 'newEdit',
createdAt: '2016-05-14 20:28:41',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10356',
pageId: 'abelian_group',
userId: 'EricRogstad',
edit: '8',
type: 'newEdit',
createdAt: '2016-05-14 20:26:39',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10283',
pageId: 'abelian_group',
userId: 'AlexeiAndreev',
edit: '7',
type: 'newEdit',
createdAt: '2016-05-14 00:30:15',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9954',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '6',
type: 'newEdit',
createdAt: '2016-05-11 00:05:40',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9953',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '5',
type: 'newEdit',
createdAt: '2016-05-11 00:02:05',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9714',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '4',
type: 'newEdit',
createdAt: '2016-05-09 07:04:40',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9712',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '3',
type: 'newEdit',
createdAt: '2016-05-09 07:02:55',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9703',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '2',
type: 'newEdit',
createdAt: '2016-05-09 06:12:33',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9702',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '1',
type: 'newEdit',
createdAt: '2016-05-09 06:11:40',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9699',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '0',
type: 'newParent',
createdAt: '2016-05-09 06:00:18',
auxPageId: 'algebraic_structure',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '9698',
pageId: 'abelian_group',
userId: 'NateSoares',
edit: '0',
type: 'newParent',
createdAt: '2016-05-09 06:00:11',
auxPageId: 'group_mathematics',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}