{
  localUrl: '../page/group_action.html',
  arbitalUrl: 'https://arbital.com/p/group_action',
  rawJsonUrl: '../raw/3t9.json',
  likeableId: '2551',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '1',
  dislikeCount: '0',
  likeScore: '1',
  individualLikes: [
    'NateSoares'
  ],
  pageId: 'group_action',
  edit: '8',
  editSummary: '',
  prevEdit: '7',
  currentEdit: '8',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Group action',
  clickbait: '"Groups, as men, will be known by their actions." ',
  textLength: '1360',
  alias: 'group_action',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-06-14 17:04:49',
  pageCreatorId: 'QiaochuYuan',
  pageCreatedAt: '2016-05-25 21:29:29',
  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: '53',
  text: 'An action of a [-3gd] $G$ on a [-3jz] $X$ is a function $\\alpha : G \\times X \\to X$ ([3vl colon-to notation]), which is often written $(g, x) \\mapsto gx$ ([3vm mapsto notation]), with $\\alpha$ omitted from the notation, such that\n\n1. $ex = x$ for all $x \\in X$, where $e$ is the identity, and\n2. $g(hx) = (gh)x$ for all $g, h \\in G, x \\in X$, where $gh$ implicitly refers to the group operation in $G$ (also omitted from the notation).\n\nEquivalently, via [currying], an action of $G$ on $X$ is a [47t group homomorphism] $G \\to \\text{Aut}(X)$, where $\\text{Aut}(X)$ is the [automorphism_group automorphism group] of $X$ (so for sets, the group of all bijections $X \\to X$, but phrasing the definition this way makes it natural to generalize to other [category_theory categories]). It's a good exercise to verify this; Arbital [49c has a proof].\n\nGroup actions are used to make precise the notion of "symmetry" in mathematics. \n\n# Examples\n\nLet $X = \\mathbb{R}^2$ be the [Euclidean_geometry Euclidean plane]. There's a group acting on $\\mathbb{R}^2$ called the [Euclidean_group Euclidean group] $ISO(2)$ which consists of all functions $f : \\mathbb{R}^2 \\to \\mathbb{R}^2$ preserving distances between two points (or equivalently all [isometry isometries]). Its elements include translations, rotations about various points, and reflections about various lines. ',
  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: '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: [
    'QiaochuYuan',
    'EricRogstad',
    'PatrickStevens'
  ],
  childIds: [
    'group_action_induces_homomorphism',
    'orbit_stabiliser_theorem',
    'stabiliser_is_a_subgroup',
    'group_orbits_partition',
    'group_stabiliser'
  ],
  parentIds: [
    'group_theory'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [
    {
      id: '3879',
      parentId: 'group_mathematics',
      childId: 'group_action',
      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: '14144',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newChild',
      createdAt: '2016-06-20 21:20:47',
      auxPageId: 'group_stabiliser',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '14087',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newChild',
      createdAt: '2016-06-20 08:55:30',
      auxPageId: 'group_orbits_partition',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '14072',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newChild',
      createdAt: '2016-06-20 08:38:38',
      auxPageId: 'stabiliser_is_a_subgroup',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13996',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newChild',
      createdAt: '2016-06-19 17:29:09',
      auxPageId: 'orbit_stabiliser_theorem',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13928',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newRequiredBy',
      createdAt: '2016-06-18 15:36:37',
      auxPageId: 'cauchy_theorem_on_subgroup_existence',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13010',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'deleteChild',
      createdAt: '2016-06-15 10:15:59',
      auxPageId: 'group_order',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13008',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '8',
      type: 'newChild',
      createdAt: '2016-06-15 10:15:52',
      auxPageId: 'group_order',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12711',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'deleteRequiredBy',
      createdAt: '2016-06-14 18:50:08',
      auxPageId: 'cayley_theorem_symmetric_groups',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12708',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '8',
      type: 'newRequirement',
      createdAt: '2016-06-14 18:48:52',
      auxPageId: 'group_mathematics',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12689',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '8',
      type: 'newEdit',
      createdAt: '2016-06-14 17:04:49',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12648',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '7',
      type: 'newChild',
      createdAt: '2016-06-14 15:47:25',
      auxPageId: 'group_action_induces_homomorphism',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12642',
      pageId: 'group_action',
      userId: 'PatrickStevens',
      edit: '7',
      type: 'newRequiredBy',
      createdAt: '2016-06-14 15:31:18',
      auxPageId: 'cayley_theorem_symmetric_groups',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '11254',
      pageId: 'group_action',
      userId: 'EricRogstad',
      edit: '7',
      type: 'newEdit',
      createdAt: '2016-05-27 20:58:51',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '11208',
      pageId: 'group_action',
      userId: 'QiaochuYuan',
      edit: '5',
      type: 'newEdit',
      createdAt: '2016-05-27 18:37:27',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '11197',
      pageId: 'group_action',
      userId: 'QiaochuYuan',
      edit: '4',
      type: 'newEdit',
      createdAt: '2016-05-27 18:19:50',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '11119',
      pageId: 'group_action',
      userId: 'EricRogstad',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-05-27 00:58:49',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '10978',
      pageId: 'group_action',
      userId: 'QiaochuYuan',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-05-25 21:29:29',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '10970',
      pageId: 'group_action',
      userId: 'QiaochuYuan',
      edit: '1',
      type: 'newParent',
      createdAt: '2016-05-25 21:22:23',
      auxPageId: 'group_theory',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'true',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}