{ localUrl: '../page/disjoint_cycles_commute_symmetric_group.html', arbitalUrl: 'https://arbital.com/p/disjoint_cycles_commute_symmetric_group', rawJsonUrl: '../raw/49g.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'disjoint_cycles_commute_symmetric_group', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'Disjoint cycles commute in symmetric groups', clickbait: 'In cycle notation, if two cycles are disjoint, then they commute.', textLength: '1366', alias: 'disjoint_cycles_commute_symmetric_group', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-14 16:53:51', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-14 16:23:56', 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: '32', text: '[summary: In a symmetric group, if we are applying a collection of permutations which are each disjoint cycles, we get the same result no matter the order in which we perform the cycles.]\n\nConsider two [49f cycles] $(a_1 a_2 \\dots a_k)$ and $(b_1 b_2 \\dots b_m)$ in the [-497] $S_n$, where all the $a_i, b_j$ are distinct.\n\nThen it is the case that the following two elements of $S_n$ are equal:\n\n- $\\sigma$, which is obtained by first performing the permutation notated by $(a_1 a_2 \\dots a_k)$ and then by performing the permutation notated by $(b_1 b_2 \\dots b_m)$\n- $\\tau$, which is obtained by first performing the permutation notated by $(b_1 b_2 \\dots b_m)$ and then by performing the permutation notated by $(a_1 a_2 \\dots a_k)$\n\nIndeed, $\\sigma(a_i) = (b_1 b_2 \\dots b_m)[(a_1 a_2 \\dots a_k)(a_i)] = (b_1 b_2 \\dots b_m)(a_{i+1}) = a_{i+1}$ (taking $a_{k+1}$ to be $a_1$), while $\\tau(a_i) = (a_1 a_2 \\dots a_k)[(b_1 b_2 \\dots b_m)(a_i)] = (a_1 a_2 \\dots a_k)(a_i) = a_{i+1}$, so they agree on elements of $(a_1 a_2 \\dots a_k)$.\nSimilarly they agree on elements of $(b_1 b_2 \\dots b_m)$; and they both do not move anything which is not an $a_i$ or a $b_j$.\nHence they are the same permutation: they act in the same way on all elements of $\\{1,2,\\dots, n\\}$.\n\nThis reasoning generalises to more than two disjoint cycles, to show that disjoint cycles commute.\n', 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: [ 'symmetric_group' ], commentIds: [], questionIds: [], tagIds: [], 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: '12688', pageId: 'disjoint_cycles_commute_symmetric_group', userId: 'PatrickStevens', edit: '2', type: 'newEdit', createdAt: '2016-06-14 16:53:51', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12663', pageId: 'disjoint_cycles_commute_symmetric_group', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-06-14 16:23:56', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12661', pageId: 'disjoint_cycles_commute_symmetric_group', userId: 'PatrickStevens', edit: '1', type: 'newParent', createdAt: '2016-06-14 16:13:18', auxPageId: 'symmetric_group', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }