{
  localUrl: '../page/left_cosets_biject.html',
  arbitalUrl: 'https://arbital.com/p/left_cosets_biject',
  rawJsonUrl: '../raw/4j8.json',
  likeableId: '2757',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '1',
  dislikeCount: '0',
  likeScore: '1',
  individualLikes: [
    'JaimeSevillaMolina'
  ],
  pageId: 'left_cosets_biject',
  edit: '2',
  editSummary: '',
  prevEdit: '1',
  currentEdit: '2',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Left cosets are all in bijection',
  clickbait: 'The left cosets of a subgroup in a parent group are all the same size.',
  textLength: '639',
  alias: 'left_cosets_biject',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-06-17 18:00:59',
  pageCreatorId: 'PatrickStevens',
  pageCreatedAt: '2016-06-17 17:47:43',
  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: '14',
  text: 'Let $H$ be a subgroup of $G$. \nThen for any two [4j4 left cosets] of $H$ in $G$, there is a [-499] between the two cosets.\n\n# Proof\n\nLet $aH, bH$ be two cosets.\nDefine the function $f: aH \\to bH$ by $x \\mapsto b a^{-1} x$.\n\nThis has the correct [3lg codomain]: if $x \\in aH$ (so $x = ah$, say), then $ba^{-1} a x = bx$ so $f(x) \\in bH$.\n\nThe function is [4b7 injective]: if $b a^{-1} x = b a^{-1} y$ then (pre-multiplying both sides by $a b^{-1}$) we obtain $x = y$.\n\nThe function is [4bg surjective]: given $b h \\in b H $, we want to find $x \\in aH$ such that $f(x) = bh$.\nLet $x = a h$ to obtain $f(x) = b a^{-1} a h = b h$, as required.',
  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: '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: [
    'group_coset'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [
    {
      id: '4510',
      parentId: 'left_cosets_biject',
      childId: 'lagrange_theorem_on_subgroup_size_intuitive',
      type: 'subject',
      creatorId: 'PatrickStevens',
      createdAt: '2016-06-28 09:30:17',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    }
  ],
  requirements: [
    {
      id: '4160',
      parentId: 'bijective_function',
      childId: 'left_cosets_biject',
      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: '14676',
      pageId: 'left_cosets_biject',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newTeacher',
      createdAt: '2016-06-28 09:30:18',
      auxPageId: 'lagrange_theorem_on_subgroup_size_intuitive',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13712',
      pageId: 'left_cosets_biject',
      userId: 'PatrickStevens',
      edit: '2',
      type: 'newRequiredBy',
      createdAt: '2016-06-17 21:24:38',
      auxPageId: 'lagrange_theorem_on_subgroup_size',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13620',
      pageId: 'left_cosets_biject',
      userId: 'PatrickStevens',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-06-17 18:00:59',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13618',
      pageId: 'left_cosets_biject',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newParent',
      createdAt: '2016-06-17 17:58:15',
      auxPageId: 'group_coset',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13607',
      pageId: 'left_cosets_biject',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-06-17 17:47:43',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13605',
      pageId: 'left_cosets_biject',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-17 17:44:16',
      auxPageId: 'bijective_function',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}