{
localUrl: '../page/orbit_stabiliser_theorem.html',
arbitalUrl: 'https://arbital.com/p/orbit_stabiliser_theorem',
rawJsonUrl: '../raw/4l8.json',
likeableId: '2921',
likeableType: 'page',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [
'EricBruylant'
],
pageId: 'orbit_stabiliser_theorem',
edit: '9',
editSummary: '',
prevEdit: '8',
currentEdit: '9',
wasPublished: 'true',
type: 'wiki',
title: 'Orbit-stabiliser theorem',
clickbait: 'The Orbit-Stabiliser theorem tells us a lot about how a group acts on a given element.',
textLength: '2981',
alias: 'orbit_stabiliser_theorem',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'MarkChimes',
editCreatedAt: '2016-07-01 04:46:57',
pageCreatorId: 'PatrickStevens',
pageCreatedAt: '2016-06-19 17:29:07',
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: '68',
text: '[summary: \nA [-3t9 group action] of a group $G$ acting on a set $X$ describes how $G$ sends elements of $X$ to other elements of $X$. Given a specific element $x \\in X$, the [-4mz stabiliser] is all those elements of the group which send $x$ back to itself, and the [-4v8 orbit] of $x$ is all the elements to which $x$ can get sent. \n\nThis theorem tells you that $G$ is divided into equal-sized pieces using $x$. Each piece "looks like" the stabilizer of $x$ (and is the same size), and the orbit of $x$ tells you how to "move the piece around" over $G$ in order to cover it. \n\nPut another way, each element $y$ in the orbit of $x$ is transformed "in the same way" by $G$ relative to $y$. \n\nThis theorem is closely related to [-4jn Lagrange's Theorem]. \n]\n\n[summary(Technical): Let $G$ be a finite [-3gd], [3t9 acting] on a set $X$. Let $x \\in X$.\nWriting $\\mathrm{Stab}_G(x)$ for the [4mz stabiliser] of $x$, and $\\mathrm{Orb}_G(x)$ for the [4v8 orbit] of $x$, we have $$|G| = |\\mathrm{Stab}_G(x)| \\times |\\mathrm{Orb}_G(x)|$$ where $| \\cdot |$ refers to the size of a set.]\n\nLet $G$ be a finite [-3gd], [3t9 acting] on a set $X$. Let $x \\in X$.\nWriting $\\mathrm{Stab}_G(x)$ for the [4mz stabiliser] of $x$, and $\\mathrm{Orb}_G(x)$ for the [4v8 orbit] of $x$, we have $$|G| = |\\mathrm{Stab}_G(x)| \\times |\\mathrm{Orb}_G(x)|$$ where $| \\cdot |$ refers to the size of a set.\n\nThis statement generalises to infinite groups, where the same proof goes through to show that there is a [499 bijection] between the [4j4 left cosets] of the group $\\mathrm{Stab}_G(x)$ and the orbit $\\mathrm{Orb}_G(x)$.\n\n# Proof\n\nRecall that the [-4lt] of the parent group.\n\nFirstly, it is enough to show that there is a bijection between the left cosets of the stabiliser, and the orbit.\nIndeed, then $$|\\mathrm{Orb}_G(x)| |\\mathrm{Stab}_G(x)| = |\\{ \\text{left cosets of} \\ \\mathrm{Stab}_G(x) \\}| |\\mathrm{Stab}_G(x)|$$\nbut the right-hand side is simply $|G|$ because an element of $G$ is specified exactly by specifying an element of the stabiliser and a coset.\n(This follows because the [4j5 cosets partition the group].)\n\n## Finding the bijection\n\nDefine $\\theta: \\mathrm{Orb}_G(x) \\to \\{ \\text{left cosets of} \\ \\mathrm{Stab}_G(x) \\}$, by $$g(x) \\mapsto g \\mathrm{Stab}_G(x)$$\n\nThis map is well-defined: note that any element of $\\mathrm{Orb}_G(x)$ is given by $g(x)$ for some $g \\in G$, so we need to show that if $g(x) = h(x)$, then $g \\mathrm{Stab}_G(x) = h \\mathrm{Stab}_G(x)$.\nThis follows: $h^{-1}g(x) = x$ so $h^{-1}g \\in \\mathrm{Stab}_G(x)$.\n\nThe map is [4b7 injective]: if $g \\mathrm{Stab}_G(x) = h \\mathrm{Stab}_G(x)$ then we need $g(x)=h(x)$.\nBut this is true: $h^{-1} g \\in \\mathrm{Stab}_G(x)$ and so $h^{-1}g(x) = x$, from which $g(x) = h(x)$.\n\nThe map is [4bg surjective]: let $g \\mathrm{Stab}_G(x)$ be a left coset.\nThen $g(x) \\in \\mathrm{Orb}_G(x)$ by definition of the orbit, so $g(x)$ gets taken to $g \\mathrm{Stab}_G(x)$ as required.\n\nHence $\\theta$ is a well-defined bijection.',
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',
'MarkChimes',
'AlexeiAndreev'
],
childIds: [
'orbit_stabiliser_theorem_external_resources'
],
parentIds: [
'group_action'
],
commentIds: [],
questionIds: [],
tagIds: [],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [
{
id: '4277',
parentId: 'group_action',
childId: 'orbit_stabiliser_theorem',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-19 17:18:00',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4278',
parentId: 'bijective_function',
childId: 'orbit_stabiliser_theorem',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-19 17:28:28',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4304',
parentId: 'stabiliser_is_a_subgroup',
childId: 'orbit_stabiliser_theorem',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-20 08:56:27',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
subjects: [],
lenses: [
{
id: '58',
pageId: 'orbit_stabiliser_theorem',
lensId: 'orbit_stabiliser_theorem_external_resources',
lensIndex: '0',
lensName: 'External resources',
lensSubtitle: '',
createdBy: '4c1',
createdAt: '2016-07-01 04:49:55',
updatedBy: '1yq',
updatedAt: '2016-07-03 00:00:18'
}
],
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: '15036',
pageId: 'orbit_stabiliser_theorem',
userId: 'MarkChimes',
edit: '0',
type: 'newChild',
createdAt: '2016-07-01 04:49:46',
auxPageId: 'orbit_stabiliser_theorem_external_resources',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15034',
pageId: 'orbit_stabiliser_theorem',
userId: 'MarkChimes',
edit: '9',
type: 'newEdit',
createdAt: '2016-07-01 04:46:57',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15033',
pageId: 'orbit_stabiliser_theorem',
userId: 'MarkChimes',
edit: '0',
type: 'deleteTag',
createdAt: '2016-07-01 04:46:02',
auxPageId: 'needs_summary_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15031',
pageId: 'orbit_stabiliser_theorem',
userId: 'MarkChimes',
edit: '8',
type: 'newEdit',
createdAt: '2016-07-01 04:45:01',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: 'Fixed some spacing in summary'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14729',
pageId: 'orbit_stabiliser_theorem',
userId: 'AlexeiAndreev',
edit: '7',
type: 'newEdit',
createdAt: '2016-06-28 22:55:44',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14726',
pageId: 'orbit_stabiliser_theorem',
userId: 'MarkChimes',
edit: '5',
type: 'newEdit',
createdAt: '2016-06-28 22:04:43',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14684',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '4',
type: 'newEdit',
createdAt: '2016-06-28 14:03:00',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14156',
pageId: 'orbit_stabiliser_theorem',
userId: 'AlexeiAndreev',
edit: '0',
type: 'newTag',
createdAt: '2016-06-20 22:04:00',
auxPageId: 'needs_summary_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14090',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '3',
type: 'newEdit',
createdAt: '2016-06-20 08:56:31',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14089',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-20 08:56:28',
auxPageId: 'stabiliser_is_a_subgroup',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14002',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '2',
type: 'newEdit',
createdAt: '2016-06-19 17:29:58',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13998',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-19 17:29:10',
auxPageId: 'group_action',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13999',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-19 17:29:10',
auxPageId: 'bijective_function',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13997',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '0',
type: 'newParent',
createdAt: '2016-06-19 17:29:09',
auxPageId: 'group_action',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13995',
pageId: 'orbit_stabiliser_theorem',
userId: 'PatrickStevens',
edit: '1',
type: 'newEdit',
createdAt: '2016-06-19 17:29:07',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'true',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}