{
  localUrl: '../page/group_isomorphism.html',
  arbitalUrl: 'https://arbital.com/p/group_isomorphism',
  rawJsonUrl: '../raw/49x.json',
  likeableId: '2759',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '1',
  dislikeCount: '0',
  likeScore: '1',
  individualLikes: [
    'EricBruylant'
  ],
  pageId: 'group_isomorphism',
  edit: '2',
  editSummary: '',
  prevEdit: '1',
  currentEdit: '2',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Group isomorphism',
  clickbait: '"Isomorphism" is the proper notion of "sameness" or "equality" among groups.',
  textLength: '1051',
  alias: 'group_isomorphism',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-06-15 08:30:30',
  pageCreatorId: 'PatrickStevens',
  pageCreatedAt: '2016-06-14 19:09:29',
  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: '33',
  text: 'A group isomorphism is a [-47t] which is [499 bijective].\nWe say that two groups are *isomorphic* if there is an isomorphism between them.\n\nIt turns out that isomorphism is a much more useful concept than true equality of [-3gd groups], and it captures the idea that "these two objects are the same group": the isomorphism shows us how to relabel the elements to see that they are indeed the same group.\n\nFor example, the trivial group is in some sense "the only group with one element", but it can be instantiated in many different ways: as $(\\{ a \\}, +_a)$, or $(\\{ b \\}, +_b)$, and so on (where $+_x$ is the [-3kb] taking $(x, x)$ to $x$).\nThey all behave in exactly the same ways for the purpose of group theory, but they are not literally identical.\nThey are all isomorphic, though: the map $\\{a \\} \\to \\{ b \\}$ given by $a \\mapsto b$ is an isomorphism of the respective groups.\n\nTwo groups are isomorphic if and only if they have the same [cayley_table Cayley table], possibly with rearrangement of rows/columns and with relabelling of elements.',
  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: '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: [
    'PatrickStevens'
  ],
  childIds: [],
  parentIds: [
    'group_mathematics',
    'isomorphism'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [
    'math2',
    'c_class_meta_tag',
    'proposed_b_class'
  ],
  relatedIds: [
    'isomorphism'
  ],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [
    {
      id: '3887',
      parentId: 'group_homomorphism',
      childId: 'group_isomorphism',
      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: '18999',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-08-20 13:15:14',
      auxPageId: 'proposed_b_class',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18998',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-08-20 13:15:09',
      auxPageId: 'c_class_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18997',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-08-20 13:13:45',
      auxPageId: 'isomorphism',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18995',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'deleteParent',
      createdAt: '2016-08-20 13:13:36',
      auxPageId: 'math',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18993',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-08-20 13:13:35',
      auxPageId: 'math2',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18992',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'deleteTag',
      createdAt: '2016-08-20 13:13:27',
      auxPageId: 'formal_definition_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '18990',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-08-20 13:13:18',
      auxPageId: 'group_mathematics',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '2760',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '1',
      dislikeCount: '0',
      likeScore: '1',
      individualLikes: [],
      id: '13799',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-06-17 23:48:58',
      auxPageId: 'math',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13751',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-06-17 23:14:49',
      auxPageId: 'formal_definition_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13748',
      pageId: 'group_isomorphism',
      userId: 'EricBruylant',
      edit: '0',
      type: 'deleteTag',
      createdAt: '2016-06-17 23:14:41',
      auxPageId: 'definition_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13232',
      pageId: 'group_isomorphism',
      userId: 'MarkChimes',
      edit: '2',
      type: 'newUsedAsTag',
      createdAt: '2016-06-16 04:02:52',
      auxPageId: 'isomorphism',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12960',
      pageId: 'group_isomorphism',
      userId: 'PatrickStevens',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-06-15 08:30:30',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12733',
      pageId: 'group_isomorphism',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newTag',
      createdAt: '2016-06-14 19:10:03',
      auxPageId: 'definition_meta_tag',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12731',
      pageId: 'group_isomorphism',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-06-14 19:09:29',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12729',
      pageId: 'group_isomorphism',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-14 19:02:27',
      auxPageId: 'group_homomorphism',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}