{
  localUrl: '../page/alternating_group_five_is_simple.html',
  arbitalUrl: 'https://arbital.com/p/alternating_group_five_is_simple',
  rawJsonUrl: '../raw/4jf.json',
  likeableId: '2746',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '1',
  dislikeCount: '0',
  likeScore: '1',
  individualLikes: [
    'EricBruylant'
  ],
  pageId: 'alternating_group_five_is_simple',
  edit: '8',
  editSummary: '',
  prevEdit: '7',
  currentEdit: '8',
  wasPublished: 'true',
  type: 'wiki',
  title: 'The alternating group on five elements is simple',
  clickbait: 'The smallest (nontrivial) simple group is the alternating group on five elements.',
  textLength: '1420',
  alias: 'alternating_group_five_is_simple',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-06-28 08:23:28',
  pageCreatorId: 'PatrickStevens',
  pageCreatedAt: '2016-06-17 18:32:38',
  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: '111',
  text: 'The [-4hf] $A_5$ on five elements is [4jc simple].\n\n# Proof\n\nRecall that $A_5$ has [3gg order] $60$, so [4jn Lagrange's theorem] states that any subgroup of $A_5$ has order dividing $60$.\n\nSuppose $H$ is a normal subgroup of $A_5$, which is not the trivial subgroup $\\{ e \\}$.\nIf $H$ has order divisible by $3$, then by [4l6 Cauchy's theorem] there is a $3$-[49f cycle] in $H$ (because the $3$-cycles are the only elements with order $3$ in $A_5$).\nBecause $H$ [4jw is a union of conjugacy classes], and because the $3$-cycles [4kv form a conjugacy class in $A_n$ for $n > 4$], $H$ would therefore contain *every* $3$-cycle; but then [4hs it would be the entire alternating group].\n\nIf instead $H$ has order divisible by $2$, then there is a double transposition such as $(12)(34)$ in $H$, since these are the only elements of order $2$ in $A_5$.\nBut then $H$ contains the entire conjugacy class so it contains every double transposition; in particular, it contains $(12)(34)$ and $(15)(34)$, so it contains $(15)(34)(12)(34) = (125)$.\nHence as before $H$ contains every $3$-cycle so is the entire alternating group.\n\nSo $H$ must have order exactly $5$, by [4jn Lagrange's theorem]; so it contains an element of order $5$ since [4jh prime order groups are cyclic].\n\nThe only such elements of $A_n$ are $5$-cycles; but the conjugacy class of a $5$-cycle is of size $12$, which is too big to fit in $H$ which has size $5$.',
  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: [
    'simplicity_of_alternating_group_five_simpler_proof'
  ],
  parentIds: [
    'alternating_group'
  ],
  commentIds: [
    '4jl'
  ],
  questionIds: [],
  tagIds: [],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [
    {
      id: '4173',
      parentId: 'simple_group',
      childId: 'alternating_group_five_is_simple',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4174',
      parentId: 'alternating_group',
      childId: 'alternating_group_five_is_simple',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4202',
      parentId: 'lagrange_theorem_on_subgroup_size',
      childId: 'alternating_group_five_is_simple',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4203',
      parentId: 'prime_order_group_is_cyclic',
      childId: 'alternating_group_five_is_simple',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4204',
      parentId: 'subgroup_normal_iff_union_of_conjugacy_classes',
      childId: 'alternating_group_five_is_simple',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4205',
      parentId: 'alternating_group_generated_by_three_cycles',
      childId: 'alternating_group_five_is_simple',
      type: 'requirement',
      creatorId: 'AlexeiAndreev',
      createdAt: '2016-06-17 21:58:56',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    },
    {
      id: '4259',
      parentId: 'cauchy_theorem_on_subgroup_existence',
      childId: 'alternating_group_five_is_simple',
      type: 'requirement',
      creatorId: 'PatrickStevens',
      createdAt: '2016-06-18 15:39:28',
      level: '1',
      isStrong: 'false',
      everPublished: 'true'
    }
  ],
  subjects: [],
  lenses: [
    {
      id: '47',
      pageId: 'alternating_group_five_is_simple',
      lensId: 'simplicity_of_alternating_group_five_simpler_proof',
      lensIndex: '0',
      lensName: 'Simpler proof',
      lensSubtitle: '',
      createdBy: '1',
      createdAt: '2016-06-17 21:58:56',
      updatedBy: '1',
      updatedAt: '2016-06-17 21:58:56'
    }
  ],
  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: '14664',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '8',
      type: 'newEdit',
      createdAt: '2016-06-28 08:23:28',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '14663',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '7',
      type: 'newEdit',
      createdAt: '2016-06-28 08:21:29',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13992',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '6',
      type: 'newEdit',
      createdAt: '2016-06-19 17:11:07',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13932',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '0',
      type: 'newRequirement',
      createdAt: '2016-06-18 15:39:28',
      auxPageId: 'cauchy_theorem_on_subgroup_existence',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13734',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '5',
      type: 'newEdit',
      createdAt: '2016-06-17 21:41:32',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13733',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '4',
      type: 'newRequirement',
      createdAt: '2016-06-17 21:41:18',
      auxPageId: 'alternating_group_generated_by_three_cycles',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13731',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '4',
      type: 'newEdit',
      createdAt: '2016-06-17 21:40:03',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13730',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '3',
      type: 'newRequirement',
      createdAt: '2016-06-17 21:39:52',
      auxPageId: 'subgroup_normal_iff_union_of_conjugacy_classes',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13728',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '3',
      type: 'newRequirement',
      createdAt: '2016-06-17 21:39:41',
      auxPageId: 'prime_order_group_is_cyclic',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13726',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '3',
      type: 'newRequirement',
      createdAt: '2016-06-17 21:39:29',
      auxPageId: 'lagrange_theorem_on_subgroup_size',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13702',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '3',
      type: 'newChild',
      createdAt: '2016-06-17 21:18:46',
      auxPageId: 'simplicity_of_alternating_group_five_simpler_proof',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13701',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '3',
      type: 'newEdit',
      createdAt: '2016-06-17 21:11:16',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13675',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-06-17 19:28:45',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13645',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '2',
      type: 'newEdit',
      createdAt: '2016-06-17 18:32:55',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13643',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-06-17 18:32:38',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13637',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-17 18:22:20',
      auxPageId: 'alternating_group',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13636',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-17 18:22:17',
      auxPageId: 'simple_group',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '13635',
      pageId: 'alternating_group_five_is_simple',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newParent',
      createdAt: '2016-06-17 18:22:10',
      auxPageId: 'alternating_group',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'true',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}