{
  localUrl: '../page/every_group_is_quotient_of_free_group.html',
  arbitalUrl: 'https://arbital.com/p/every_group_is_quotient_of_free_group',
  rawJsonUrl: '../raw/5jb.json',
  likeableId: '3199',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '1',
  dislikeCount: '0',
  likeScore: '1',
  individualLikes: [
    'EricBruylant'
  ],
  pageId: 'every_group_is_quotient_of_free_group',
  edit: '1',
  editSummary: '',
  prevEdit: '0',
  currentEdit: '1',
  wasPublished: 'true',
  type: 'wiki',
  title: 'Every group is a quotient of a free group',
  clickbait: '',
  textLength: '1243',
  alias: 'every_group_is_quotient_of_free_group',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-07-22 11:38:50',
  pageCreatorId: 'PatrickStevens',
  pageCreatedAt: '2016-07-22 11:38:50',
  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: '29',
  text: 'Given a [3gd group] $G$, there is a [-free_group] $F(X)$ on some set $X$, such that $G$ is [49x isomorphic] to some [4tq quotient] of $F(X)$.\n\nThis is an instance of a much more general phenomenon: for a general [monad_category_theory monad] $T: \\mathcal{C} \\to \\mathcal{C}$ where $\\mathcal{C}$ is a category, if $(A, \\alpha)$ is an [eilenberg_moore_category algebra] over $T$, then $\\alpha: TA \\to A$ is a [coequaliser_category_theory coequaliser]. ([algebras_are_coequalisers Proof.])\n\n# Proof\nLet $F(G)$ be the free group on the elements of $G$, in a slight abuse of notation where we use $G$ interchangeably with its [-3gz].\nDefine the [47t homomorphism] $\\theta: F(G) \\to G$ by "multiplying out a word": taking the word $(a_1, a_2, \\dots, a_n)$ to the product $a_1 a_2 \\dots a_n$.\n\nThis is indeed a group homomorphism, because the group operation in $F(G)$ is concatenation and the group operation in $G$ is multiplication: clearly if $w_1 = (a_1, \\dots, a_m)$, $w_2 = (b_1, \\dots, b_n)$ are words, then $$\\theta(w_1 w_2) = \\theta(a_1, \\dots, a_m, b_1, \\dots, b_m) = a_1 \\dots a_m b_1 \\dots b_m = \\theta(w_1) \\theta(w_2)$$\n\nThis immediately expresses $G$ as a quotient of $F(G)$, since [4h7 kernels of homomorphisms are normal subgroups].',
  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: [
    'group_mathematics'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [
    'math3'
  ],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [],
  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: '17294',
      pageId: 'every_group_is_quotient_of_free_group',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newParent',
      createdAt: '2016-07-22 11:38:52',
      auxPageId: 'group_mathematics',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '3194',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '1',
      dislikeCount: '0',
      likeScore: '1',
      individualLikes: [],
      id: '17295',
      pageId: 'every_group_is_quotient_of_free_group',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newTag',
      createdAt: '2016-07-22 11:38:52',
      auxPageId: 'math3',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '3183',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '1',
      dislikeCount: '0',
      likeScore: '1',
      individualLikes: [],
      id: '17292',
      pageId: 'every_group_is_quotient_of_free_group',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-07-22 11:38:50',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}