{
  localUrl: '../page/even_signed_permutations_form_a_group.html',
  arbitalUrl: 'https://arbital.com/p/even_signed_permutations_form_a_group',
  rawJsonUrl: '../raw/4hg.json',
  likeableId: '0',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '0',
  dislikeCount: '0',
  likeScore: '0',
  individualLikes: [],
  pageId: 'even_signed_permutations_form_a_group',
  edit: '1',
  editSummary: '',
  prevEdit: '0',
  currentEdit: '1',
  wasPublished: 'true',
  type: 'wiki',
  title: 'The collection of even-signed permutations is a group',
  clickbait: 'This proves the well-definedness of one particular definition of the alternating group.',
  textLength: '1041',
  alias: 'even_signed_permutations_form_a_group',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-06-17 13:43:40',
  pageCreatorId: 'PatrickStevens',
  pageCreatedAt: '2016-06-17 13:43:40',
  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: '33',
  text: 'The collection of elements of the [-497] $S_n$ which are made by multiplying together an even number of permutations forms a subgroup of $S_n$.\n\nThis proves that the [-alternating_group] $A_n$ is well-defined, if it is given as "the subgroup of $S_n$ containing precisely that which is made by multiplying together an even number of transpositions".\n\n# Proof\n\nFirstly we must check that "I can only be made by multiplying together an even number of transpositions" is a well-defined notion; this [4hh is in fact true].\n\nWe must check the group axioms.\n\n- Identity: the identity is simply the product of no transpositions, and $0$ is even.\n- Associativity is inherited from $S_n$.\n- Closure: if we multiply together an even number of transpositions, and then a further even number of transpositions, we obtain an even number of transpositions.\n- Inverses: if $\\sigma$ is made of an even number of transpositions, say $\\tau_1 \\tau_2 \\dots \\tau_m$, then its inverse is $\\tau_m \\tau_{m-1} \\dots \\tau_1$, since a transposition is its own inverse.',
  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: '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: [
    'alternating_group'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [
    {
      id: '4117',
      parentId: 'transposition_in_symmetric_group',
      childId: 'even_signed_permutations_form_a_group',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4118',
      parentId: 'symmetric_group',
      childId: 'even_signed_permutations_form_a_group',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4124',
      parentId: 'sign_of_permutation_is_well_defined',
      childId: 'even_signed_permutations_form_a_group',
      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: '13517',
      pageId: 'even_signed_permutations_form_a_group',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newParent',
      createdAt: '2016-06-17 14:14:21',
      auxPageId: 'alternating_group',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13501',
      pageId: 'even_signed_permutations_form_a_group',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-06-17 13:43:40',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13497',
      pageId: 'even_signed_permutations_form_a_group',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-17 13:43:39',
      auxPageId: 'sign_of_permutation_is_well_defined',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13486',
      pageId: 'even_signed_permutations_form_a_group',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-17 13:16:55',
      auxPageId: 'symmetric_group',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13485',
      pageId: 'even_signed_permutations_form_a_group',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-17 13:16:47',
      auxPageId: 'transposition_in_symmetric_group',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}