{ localUrl: '../page/group_action.html', arbitalUrl: 'https://arbital.com/p/group_action', rawJsonUrl: '../raw/3t9.json', likeableId: '2551', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'NateSoares' ], pageId: 'group_action', edit: '8', editSummary: '', prevEdit: '7', currentEdit: '8', wasPublished: 'true', type: 'wiki', title: 'Group action', clickbait: '"Groups, as men, will be known by their actions." ', textLength: '1360', alias: 'group_action', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-14 17:04:49', pageCreatorId: 'QiaochuYuan', pageCreatedAt: '2016-05-25 21:29:29', 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: '39', text: 'An action of a [-3gd] $G$ on a [-3jz] $X$ is a function $\\alpha : G \\times X \\to X$ ([3vl colon-to notation]), which is often written $(g, x) \\mapsto gx$ ([3vm mapsto notation]), with $\\alpha$ omitted from the notation, such that\n\n1. $ex = x$ for all $x \\in X$, where $e$ is the identity, and\n2. $g(hx) = (gh)x$ for all $g, h \\in G, x \\in X$, where $gh$ implicitly refers to the group operation in $G$ (also omitted from the notation).\n\nEquivalently, via [currying], an action of $G$ on $X$ is a [47t group homomorphism] $G \\to \\text{Aut}(X)$, where $\\text{Aut}(X)$ is the [automorphism_group automorphism group] of $X$ (so for sets, the group of all bijections $X \\to X$, but phrasing the definition this way makes it natural to generalize to other [category_theory categories]). It's a good exercise to verify this; Arbital [49c has a proof].\n\nGroup actions are used to make precise the notion of "symmetry" in mathematics. \n\n# Examples\n\nLet $X = \\mathbb{R}^2$ be the [Euclidean_geometry Euclidean plane]. There's a group acting on $\\mathbb{R}^2$ called the [Euclidean_group Euclidean group] $ISO(2)$ which consists of all functions $f : \\mathbb{R}^2 \\to \\mathbb{R}^2$ preserving distances between two points (or equivalently all [isometry isometries]). Its elements include translations, rotations about various points, and reflections about various lines. ', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', 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: [ 'QiaochuYuan', 'EricRogstad', 'PatrickStevens' ], childIds: [ 'group_action_induces_homomorphism', 'orbit_stabiliser_theorem', 'stabiliser_is_a_subgroup', 'group_orbits_partition', 'group_stabiliser' ], parentIds: [ 'group_theory' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [ { id: '3879', parentId: 'group_mathematics', childId: 'group_action', 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: '14144', pageId: 'group_action', userId: 'PatrickStevens', edit: '0', type: 'newChild', createdAt: '2016-06-20 21:20:47', auxPageId: 'group_stabiliser', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '14087', pageId: 'group_action', userId: 'PatrickStevens', edit: '0', type: 'newChild', createdAt: '2016-06-20 08:55:30', auxPageId: 'group_orbits_partition', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '14072', pageId: 'group_action', userId: 'PatrickStevens', edit: '0', type: 'newChild', createdAt: '2016-06-20 08:38:38', auxPageId: 'stabiliser_is_a_subgroup', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13996', pageId: 'group_action', userId: 'PatrickStevens', edit: '0', type: 'newChild', createdAt: '2016-06-19 17:29:09', auxPageId: 'orbit_stabiliser_theorem', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13928', pageId: 'group_action', userId: 'PatrickStevens', edit: '0', type: 'newRequiredBy', createdAt: '2016-06-18 15:36:37', auxPageId: 'cauchy_theorem_on_subgroup_existence', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13010', pageId: 'group_action', userId: 'PatrickStevens', edit: '0', type: 'deleteChild', createdAt: '2016-06-15 10:15:59', auxPageId: 'group_order', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13008', pageId: 'group_action', userId: 'PatrickStevens', edit: '8', type: 'newChild', createdAt: '2016-06-15 10:15:52', auxPageId: 'group_order', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12711', pageId: 'group_action', userId: 'PatrickStevens', edit: '0', type: 'deleteRequiredBy', createdAt: '2016-06-14 18:50:08', auxPageId: 'cayley_theorem_symmetric_groups', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12708', pageId: 'group_action', userId: 'PatrickStevens', edit: '8', type: 'newRequirement', createdAt: '2016-06-14 18:48:52', auxPageId: 'group_mathematics', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12689', pageId: 'group_action', userId: 'PatrickStevens', edit: '8', type: 'newEdit', createdAt: '2016-06-14 17:04:49', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12648', pageId: 'group_action', userId: 'PatrickStevens', edit: '7', type: 'newChild', createdAt: '2016-06-14 15:47:25', auxPageId: 'group_action_induces_homomorphism', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '12642', pageId: 'group_action', userId: 'PatrickStevens', edit: '7', type: 'newRequiredBy', createdAt: '2016-06-14 15:31:18', auxPageId: 'cayley_theorem_symmetric_groups', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11254', pageId: 'group_action', userId: 'EricRogstad', edit: '7', type: 'newEdit', createdAt: '2016-05-27 20:58:51', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11208', pageId: 'group_action', userId: 'QiaochuYuan', edit: '5', type: 'newEdit', createdAt: '2016-05-27 18:37:27', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11197', pageId: 'group_action', userId: 'QiaochuYuan', edit: '4', type: 'newEdit', createdAt: '2016-05-27 18:19:50', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '11119', pageId: 'group_action', userId: 'EricRogstad', edit: '2', type: 'newEdit', createdAt: '2016-05-27 00:58:49', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10978', pageId: 'group_action', userId: 'QiaochuYuan', edit: '1', type: 'newEdit', createdAt: '2016-05-25 21:29:29', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10970', pageId: 'group_action', userId: 'QiaochuYuan', edit: '1', type: 'newParent', createdAt: '2016-05-25 21:22:23', auxPageId: 'group_theory', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'true', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }