{ localUrl: '../page/group_coset.html', arbitalUrl: 'https://arbital.com/p/group_coset', rawJsonUrl: '../raw/4j4.json', likeableId: '2754', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'group_coset', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Group coset', clickbait: '', textLength: '1874', alias: 'group_coset', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-17 17:58:15', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-17 17:58:15', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '2', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '41', text: 'Given a subgroup $H$ of [-3gd] $G$, the *left cosets* of $H$ in $G$ are sets of the form $\\{ gh : h \\in H \\}$, for some $g \\in G$.\nThis is written $gH$ as a shorthand.\n\nSimilarly, the *right cosets* are the sets of the form $Hg = \\{ hg: h \\in H \\}$.\n\n# Examples\n%%%knows-requisite([497]):\n## Symmetric group\n\nIn $S_3$, the [-497] on three elements, we can list the elements as $\\{ e, (123), (132), (12), (13), (23) \\}$, using [49f cycle notation].\nDefine $A_3$ (which happens to have a name: the [-4hf]) to be the subgroup with elements $\\{ e, (123), (132) \\}$.\n\nThen the coset $(12) A_3$ has elements $\\{ (12), (12)(123), (12)(132) \\}$, which is simplified to $\\{ (12), (23), (13) \\}$.\n\nThe coset $(123)A_3$ is simply $A_3$, because $A_3$ is a subgroup so is closed under the group operation. $(123)$ is already in $A_3$.\n%%%\n\n[todo: more examples, with different requirements]\n\n# Properties\n\n- The left cosets of $H$ in $G$ [set_partition partition] $G$. ([4j5 Proof.])\n- For any pair of left cosets of $H$, there is a [499 bijection] between them; that is, all the cosets are all the same size. ([4j8 Proof.])\n\n# Why are we interested in cosets?\n\nUnder certain conditions (namely that the subgroup $H$ must be [4h6 normal]), we may define the [-quotient_group], a very important concept; see the page on [4j5 "left cosets partition the parent group"] for a glance at why this is useful.\n[todo: there must be a less clumsy way to do it]\n\nAdditionally, there is a key theorem whose usual proof considers cosets ([lagrange_theorem_on_subgroup_size Lagrange's theorem]) which strongly restricts the possible sizes of subgroups of $G$, and which itself is enough to classify all the groups of [3gg order] $p$ for $p$ [prime_number prime].\nLagrange's theorem also has very common applications in [-number_theory], in the form of the [fermat_euler_theorem Fermat-Euler theorem].', 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: 'true', 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: [ 'left_cosets_partition_parent_group', 'left_cosets_biject' ], parentIds: [ 'group_mathematics' ], commentIds: [], questionIds: [], tagIds: [ 'formal_definition_meta_tag', 'needs_clickbait_meta_tag' ], 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: '17120', pageId: 'group_coset', userId: 'EricBruylant', edit: '0', type: 'deleteTag', createdAt: '2016-07-19 02:02:53', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17118', pageId: 'group_coset', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-19 02:02:52', auxPageId: 'formal_definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17117', pageId: 'group_coset', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-19 02:02:45', auxPageId: 'needs_clickbait_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13710', pageId: 'group_coset', userId: 'PatrickStevens', edit: '1', type: 'newRequiredBy', createdAt: '2016-06-17 21:24:38', auxPageId: 'lagrange_theorem_on_subgroup_size', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13621', pageId: 'group_coset', userId: 'PatrickStevens', edit: '1', type: 'newRequiredBy', createdAt: '2016-06-17 18:03:28', auxPageId: 'left_cosets_partition_parent_group', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13619', pageId: 'group_coset', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-06-17 17:58:15', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13590', pageId: 'group_coset', userId: 'PatrickStevens', edit: '1', type: 'newTag', createdAt: '2016-06-17 17:19:09', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13589', pageId: 'group_coset', userId: 'PatrickStevens', edit: '1', type: 'newParent', createdAt: '2016-06-17 17:19:05', auxPageId: 'group_mathematics', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'true', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }