{ 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: '21', 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: {} }