{ localUrl: '../page/group_conjugate.html', arbitalUrl: 'https://arbital.com/p/group_conjugate', rawJsonUrl: '../raw/4gk.json', likeableId: '2724', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'group_conjugate', edit: '6', editSummary: '', prevEdit: '5', currentEdit: '6', wasPublished: 'true', type: 'wiki', title: 'Group conjugate', clickbait: 'Conjugation lets us perform permutations "from the point of view of" another permutation.', textLength: '2674', alias: 'group_conjugate', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-20 09:05:54', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-16 20:48:36', 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: '44', text: 'Two elements $x, y$ of a [-3gd] $G$ are *conjugate* if there is some $h \\in G$ such that $hxh^{-1} = y$.\n\n# Conjugacy as "changing the worldview"\n\nConjugating by $h$ is equivalent to "viewing the world through $h$'s eyes".\nThis is most easily demonstrated in the [-497], where it [4bh is a fact] that if $$\\sigma = (a_{11} a_{12} \\dots a_{1 n_1})(a_{21} \\dots a_{2 n_2}) \\dots (a_{k 1} a_{k 2} \\dots a_{k n_k})$$\nand $\\tau \\in S_n$, then $$\\tau \\sigma \\tau^{-1} = (\\tau(a_{11}) \\tau(a_{12}) \\dots \\tau(a_{1 n_1}))(\\tau(a_{21}) \\dots \\tau(a_{2 n_2})) \\dots (\\tau(a_{k 1}) \\tau(a_{k 2}) \\dots \\tau(a_{k n_k}))$$\n\nThat is, conjugating by $\\tau$ has "caused us to view $\\sigma$ from the point of view of $\\tau$".\n\nSimilarly, in the [-4cy] $D_{2n}$ on $n$ vertices, conjugation of the rotation by a reflection yields the inverse of the rotation: it is "the rotation, but viewed as acting on the reflected polygon".\nEquivalently, if the polygon is sitting on a glass table, conjugating the rotation by a reflection makes the rotation act "as if we had moved our head under the table to look upwards first".\n\nIn general, if $G$ is a group which [3t9 acts] as (some of) the symmetries of a certain object $X$ %%note:Which [49b we can always view as being the case].%% then conjugation of $g \\in G$ by $h \\in G$ produces a symmetry $hgh^{-1}$ which acts in the same way as $g$ does, but on a copy of $X$ which has already been permuted by $h$.\n\n# Closure under conjugation\n\nIf a subgroup $H$ of $G$ is closed under conjugation by elements of $G$, then $H$ is a [-4h6].\nThe concept of a normal subgroup is extremely important in group theory.\n\n%%%knows-requisite([3t9]):\n# Conjugation action\n\nConjugation forms a [3t9 action].\nFormally, let $G$ act on itself: $\\rho: G \\times G \\to G$, with $\\rho(g, k) = g k g^{-1}$.\nIt is an exercise to show that this is indeed an action.\n%%hidden(Show solution):\nWe need to show that the identity acts trivially, and that products may be broken up to act individually.\n\n- $\\rho(gh, k) = (gh)k(gh)^{-1} = ghkh^{-1}g^{-1} = g \\rho(h, k) g^{-1} = \\rho(g, \\rho(h, k))$;\n- $\\rho(e, k) = eke^{-1} = k$.\n%%\n\nThe [group_stabiliser stabiliser] of this action, $\\mathrm{Stab}_G(g)$ for some fixed $g \\in G$, is the set of all elements such that $kgk^{-1} = g$: that is, such that $kg = gk$.\nEquivalently, it is the [group_centraliser centraliser] of $g$ in $G$: it is the subgroup of all elements which commute with $G$.\n\nThe [group_orbit orbit] of the action, $\\mathrm{Orb}_G(g)$ for some fixed $g \\in G$, is the [-4bj] of $g$ in $G$.\nBy the [-4l8], this immediately gives that the size of a conjugacy class divides the [3gg order] of the parent group.\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: '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: [], parentIds: [ 'group_mathematics' ], commentIds: [], questionIds: [], tagIds: [ 'math2' ], 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: '18988', pageId: 'group_conjugate', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-08-20 13:11:24', auxPageId: 'math2', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18987', pageId: 'group_conjugate', userId: 'EricBruylant', edit: '0', type: 'deleteParent', createdAt: '2016-08-20 13:10:51', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18985', pageId: 'group_conjugate', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-08-20 13:10:48', auxPageId: 'group_mathematics', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '14092', pageId: 'group_conjugate', userId: 'PatrickStevens', edit: '6', type: 'newEdit', createdAt: '2016-06-20 09:05:54', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13538', pageId: 'group_conjugate', userId: 'PatrickStevens', edit: '5', type: 'newEdit', createdAt: '2016-06-17 15:16:15', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13467', pageId: 'group_conjugate', userId: 'PatrickStevens', edit: '4', type: 'newRequiredBy', createdAt: '2016-06-17 10:27:44', auxPageId: 'normal_subgroup', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '2728', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '13392', pageId: 'group_conjugate', userId: 'EricBruylant', edit: '4', type: 'newParent', createdAt: '2016-06-16 21:02:20', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13389', pageId: 'group_conjugate', userId: 'PatrickStevens', edit: '4', type: 'newEdit', createdAt: '2016-06-16 20:51:34', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13388', pageId: 'group_conjugate', userId: 'PatrickStevens', edit: '3', type: 'newEdit', createdAt: '2016-06-16 20:49:24', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13385', pageId: 'group_conjugate', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-06-16 20:48:36', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }