{
localUrl: '../page/group_examples.html',
arbitalUrl: 'https://arbital.com/p/group_examples',
rawJsonUrl: '../raw/3t1.json',
likeableId: '0',
likeableType: 'page',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
pageId: 'group_examples',
edit: '11',
editSummary: '',
prevEdit: '9',
currentEdit: '11',
wasPublished: 'true',
type: 'wiki',
title: 'Group: Examples',
clickbait: 'Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today? ',
textLength: '1570',
alias: 'group_examples',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'PatrickStevens',
editCreatedAt: '2016-10-21 17:25:45',
pageCreatorId: 'QiaochuYuan',
pageCreatedAt: '2016-05-25 20:45:31',
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: '165',
text: '[summary:\nExamples of [-3gd groups], including the [-497 symmetric groups] and [-general_linear_group general linear groups].\n]\n\n# The symmetric groups\n\nFor every positive integer $n$ there is a group $S_n$, the [497 symmetric group] of order $n$, defined as the group of all permutations (bijections) $\\{ 1, 2, \\dots n \\} \\to \\{ 1, 2, \\dots n \\}$ (or any other [-3jz] with $n$ elements). The symmetric groups play a central role in group theory: for example, a [3t9 group action] of a group $G$ on a set $X$ with $n$ elements is the same as a [47t homomorphism] $G \\to S_n$. \n\nUp to [4bj conjugacy], a permutation is determined by its [4cg cycle type]. \n\n# The dihedral groups\n\nThe [4cy dihedral groups] $D_{2n}$ are the collections of symmetries of an $n$-sided regular polygon. It has a [5j9 presentation] $\\langle r, f \\mid r^n, f^2, (rf)^2 \\rangle$, where $r$ represents rotation by $\\tau/n$ degrees, and $f$ represents reflection. \n\nFor $n > 2$, the dihedral groups are non-commutative.\n\n# The general linear groups\n\nFor every [481 field] $K$ and positive integer $n$ there is a group $GL_n(K)$, the [general_linear_group general linear group] of order $n$ over $K$. Concretely, this is the group of all invertible $n \\times n$ [matrix matrices] with entries in $K$; more abstractly, this is the [automorphism automorphism group] of a [3w0 vector space] of [vector_space_dimension dimension] $n$ over $K$. \n\nIf $K$ is [algebraically_closed_field algebraically closed], then up to conjugacy, a matrix is determined by its [Jordan_normal_form Jordan normal form]. ',
metaText: '',
isTextLoaded: 'true',
isSubscribedToDiscussion: 'false',
isSubscribedToUser: 'false',
isSubscribedAsMaintainer: 'false',
discussionSubscriberCount: '1',
maintainerCount: '1',
userSubscriberCount: '0',
lastVisit: '',
hasDraft: 'false',
votes: [],
voteSummary: [
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0',
'0'
],
muVoteSummary: '0',
voteScaling: '0',
currentUserVote: '-2',
voteCount: '0',
lockedVoteType: '',
maxEditEver: '0',
redLinkCount: '0',
lockedBy: '',
lockedUntil: '',
nextPageId: '',
prevPageId: '',
usedAsMastery: 'false',
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: [
'QiaochuYuan',
'PatrickStevens',
'EricBruylant',
'MarkChimes',
'DanielSatanove'
],
childIds: [],
parentIds: [
'group_mathematics'
],
commentIds: [],
questionIds: [],
tagIds: [],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [],
subjects: [],
lenses: [],
lensParentId: 'group_mathematics',
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: '20230',
pageId: 'group_examples',
userId: 'PatrickStevens',
edit: '11',
type: 'newEdit',
createdAt: '2016-10-21 17:25:45',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '20195',
pageId: 'group_examples',
userId: 'DanielSatanove',
edit: '9',
type: 'newEdit',
createdAt: '2016-10-20 23:50:20',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15020',
pageId: 'group_examples',
userId: 'MarkChimes',
edit: '0',
type: 'deleteTag',
createdAt: '2016-07-01 03:17:21',
auxPageId: 'needs_summary_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15018',
pageId: 'group_examples',
userId: 'MarkChimes',
edit: '8',
type: 'newEdit',
createdAt: '2016-07-01 03:16:59',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14399',
pageId: 'group_examples',
userId: 'EricBruylant',
edit: '7',
type: 'newEdit',
createdAt: '2016-06-22 17:56:35',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14398',
pageId: 'group_examples',
userId: 'EricBruylant',
edit: '0',
type: 'newTag',
createdAt: '2016-06-22 17:56:21',
auxPageId: 'needs_summary_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14397',
pageId: 'group_examples',
userId: 'EricBruylant',
edit: '0',
type: 'deleteTag',
createdAt: '2016-06-22 17:56:04',
auxPageId: 'needs_technical_summary_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14395',
pageId: 'group_examples',
userId: 'EricBruylant',
edit: '0',
type: 'newTag',
createdAt: '2016-06-22 17:56:01',
auxPageId: 'needs_technical_summary_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12633',
pageId: 'group_examples',
userId: 'PatrickStevens',
edit: '6',
type: 'newEdit',
createdAt: '2016-06-14 12:34:04',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12543',
pageId: 'group_examples',
userId: 'PatrickStevens',
edit: '5',
type: 'newEdit',
createdAt: '2016-06-13 16:10:40',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10994',
pageId: 'group_examples',
userId: 'QiaochuYuan',
edit: '4',
type: 'newEdit',
createdAt: '2016-05-25 21:47:57',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10945',
pageId: 'group_examples',
userId: 'QiaochuYuan',
edit: '0',
type: 'newAlias',
createdAt: '2016-05-25 21:04:35',
auxPageId: '',
oldSettingsValue: 'groups_by_example',
newSettingsValue: 'group_examples'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10946',
pageId: 'group_examples',
userId: 'QiaochuYuan',
edit: '3',
type: 'newEdit',
createdAt: '2016-05-25 21:04:35',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10933',
pageId: 'group_examples',
userId: 'QiaochuYuan',
edit: '2',
type: 'newEdit',
createdAt: '2016-05-25 20:47:24',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10930',
pageId: 'group_examples',
userId: 'QiaochuYuan',
edit: '1',
type: 'newEdit',
createdAt: '2016-05-25 20:45:31',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '10921',
pageId: 'group_examples',
userId: 'QiaochuYuan',
edit: '1',
type: 'newParent',
createdAt: '2016-05-25 20:41:34',
auxPageId: 'group_mathematics',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}