{
  localUrl: '../page/image_of_group_under_homomorphism_is_subgroup.html',
  arbitalUrl: 'https://arbital.com/p/image_of_group_under_homomorphism_is_subgroup',
  rawJsonUrl: '../raw/4b4.json',
  likeableId: '0',
  likeableType: 'page',
  myLikeValue: '0',
  likeCount: '0',
  dislikeCount: '0',
  likeScore: '0',
  individualLikes: [],
  pageId: 'image_of_group_under_homomorphism_is_subgroup',
  edit: '1',
  editSummary: '',
  prevEdit: '0',
  currentEdit: '1',
  wasPublished: 'true',
  type: 'wiki',
  title: 'The image of a group under a homomorphism is a subgroup of the codomain',
  clickbait: 'Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.',
  textLength: '903',
  alias: 'image_of_group_under_homomorphism_is_subgroup',
  externalUrl: '',
  sortChildrenBy: 'likes',
  hasVote: 'false',
  voteType: '',
  votesAnonymous: 'false',
  editCreatorId: 'PatrickStevens',
  editCreatedAt: '2016-06-14 19:30:27',
  pageCreatorId: 'PatrickStevens',
  pageCreatedAt: '2016-06-14 19:30:27',
  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: '16',
  text: 'Let $f: G \\to H$ be a [-47t], and write $f(G)$ for the set $\\{ f(g) : g \\in G \\}$.\nThen $f(G)$ is a group under the operation inherited from $H$.\n\n# Proof\n\nTo prove this, we must verify the group axioms.\nLet $f: G \\to H$ be a group homomorphism, and let $e_G, e_H$ be the identities of $G$ and of $H$ respectively.\nWrite $f(G)$ for the image of $G$.\n\nThen $f(G)$ is closed under the operation of $H$: since $f(g) f(h) = f(gh)$, so the result of $H$-multiplying two elements of $f(G)$ is also in $f(G)$.\n\n$e_H$ is the identity for $f(G)$: it is $f(e_G)$, so it does lie in the image, while it acts as the identity because $f(e_G) f(g) = f(e_G g) = f(g)$, and likewise for multiplication on the right.\n\nInverses exist, by "the inverse of the image is the image of the inverse".\n\nThe operation remains associative: this is inherited from $H$.\n\nTherefore, $f(G)$ is a group, and indeed is a subgroup of $H$.',
  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_homomorphism'
  ],
  commentIds: [],
  questionIds: [],
  tagIds: [],
  relatedIds: [],
  markIds: [],
  explanations: [],
  learnMore: [],
  requirements: [
    {
      id: '3902',
      parentId: 'group_homomorphism',
      childId: 'image_of_group_under_homomorphism_is_subgroup',
      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: '12765',
      pageId: 'image_of_group_under_homomorphism_is_subgroup',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newEdit',
      createdAt: '2016-06-14 19:30:27',
      auxPageId: '',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12762',
      pageId: 'image_of_group_under_homomorphism_is_subgroup',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newRequirement',
      createdAt: '2016-06-14 19:27:50',
      auxPageId: 'group_homomorphism',
      oldSettingsValue: '',
      newSettingsValue: ''
    },
    {
      likeableId: '0',
      likeableType: 'changeLog',
      myLikeValue: '0',
      likeCount: '0',
      dislikeCount: '0',
      likeScore: '0',
      individualLikes: [],
      id: '12761',
      pageId: 'image_of_group_under_homomorphism_is_subgroup',
      userId: 'PatrickStevens',
      edit: '1',
      type: 'newParent',
      createdAt: '2016-06-14 19:26:56',
      auxPageId: 'group_homomorphism',
      oldSettingsValue: '',
      newSettingsValue: ''
    }
  ],
  feedSubmissions: [],
  searchStrings: {},
  hasChildren: 'false',
  hasParents: 'true',
  redAliases: {},
  improvementTagIds: [],
  nonMetaTagIds: [],
  todos: [],
  slowDownMap: 'null',
  speedUpMap: 'null',
  arcPageIds: 'null',
  contentRequests: {}
}