{
localUrl: '../page/lagrange_theorem_on_subgroup_size_intuitive.html',
arbitalUrl: 'https://arbital.com/p/lagrange_theorem_on_subgroup_size_intuitive',
rawJsonUrl: '../raw/4vz.json',
likeableId: '2868',
likeableType: 'page',
myLikeValue: '0',
likeCount: '2',
dislikeCount: '0',
likeScore: '2',
individualLikes: [
'EricBruylant',
'PatrickStevens'
],
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
edit: '9',
editSummary: 'updated link to cardinality',
prevEdit: '8',
currentEdit: '9',
wasPublished: 'true',
type: 'wiki',
title: 'Lagrange theorem on subgroup size: Intuitive version',
clickbait: 'Lagrange's theorem strongly restricts the size a subgroup of a group can be.',
textLength: '4324',
alias: 'lagrange_theorem_on_subgroup_size_intuitive',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'EricBruylant',
editCreatedAt: '2016-08-27 14:11:30',
pageCreatorId: 'PatrickStevens',
pageCreatedAt: '2016-06-28 09:00:01',
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: '50',
text: '[summary: Lagrange's Theorem gives us a restriction on the size of a subgroup of a finite group: namely, subgroups have size dividing the parent group.]\n\nGiven a finite [-3gd] $G$, it may have many [576 subgroups].\nSo far, we know almost nothing about those subgroups; it would be great if we had some way of restricting them.\n\nAn example of such a restriction, which we do already know, is that a subgroup $H$ of $G$ has to have [3gg size] less than or equal to the size of $G$ itself.\nThis is because $H$ is contained in $G$, and if the set $X$ is contained in the set $Y$ then the size of $X$ is less than or equal to the size of $Y$.\n(This would have to be true for any reasonable definition of "size"; the [4w5 usual definition] certainly has this property.)\n\nLagrange's Theorem gives us a much more powerful restriction: not only is the size $|H|$ of $H$ less than or equal to $|G|$, but in fact $|H|$ divides $|G|$.\n\n%%hidden(Example: subgroups of the cyclic group on six elements):\n*A priori*, all we know about the subgroups of the [-47y] $C_6$ of order $6$ is that they are of order $1, 2, 3, 4, 5$ or $6$.\n\nLagrange's Theorem tells us that they can only be of order $1, 2, 3$ or $6$: there are no subgroups of order $4$ or $5$.\nLagrange tells us nothing about whether there *are* subgroups of size $1,2,3$ or $6$: only that if we are given a subgroup, then it is of one of those sizes.\n\nIn fact, as an aside, there are indeed subgroups of sizes $1,2,3,6$:\n\n- the subgroup containing only the identity is of order $1$\n- the "improper" subgroup $C_6$ is of order $6$\n- subgroups of size $2$ and $3$ are guaranteed by [4l6 Cauchy's theorem].\n\n%%\n\n# Proof\n\nIn order to show that $|H|$ divides $|G|$, we would be done if we could divide the elements of $G$ up into separate buckets of size $|H|$.\n\nThere is a fairly obvious place to start: we already have one bucket of size $|H|$, namely $H$ itself (which consists of some elements of $G$).\nCan we perhaps use this to create more buckets of size $|H|$?\n\nFor motivation: if we think of $H$ as being a collection of symmetries (which we can do, by [49b Cayley's Theorem] which states that all groups may be viewed as collections of symmetries), then we can create more symmetries by "tacking on elements of $G$".\n\nFormally, let $g$ be an element of $G$, and consider $gH = \\{ g h : h \\in H \\}$.\n\nExercise: every element of $G$ does have one of these buckets $gH$ in which it lies.\n%%hidden(Show solution):\nThe element $g$ of $G$ is contained in the bucket $gH$, because the identity $e$ is contained in $H$ and so $ge$ is in $gH$; but $ge = g$.\n%%\n\nExercise: $gH$ is a set of size $|H|$. %%note:More formally put, [-4j8].%%\n%%hidden(Show solution):\nIn order to show that $gH$ has size $|H|$, it is enough to match up the elements of $gH$ [499 bijectively] with the elements of $|H|$.\n\nWe can do this with the [-3jy] $H \\to gH$ taking $h \\in H$ and producing $gh$.\nThis has an [4sn inverse]: the function $gH \\to H$ which is given by pre-multiplying by $g^{-1}$, so that $gx \\mapsto g^{-1} g x = x$.\n%%\n\nNow, are all these buckets separate? Do any of them overlap?\n\nExercise: if $x \\in rH$ and $x \\in sH$ then $rH = sH$. That is, if any two buckets intersect then they are the same bucket. %%note:More formally put, [4j5].%%\n%%hidden(Show solution):\nSuppose $x \\in rH$ and $x \\in sH$.\n\nThen $x = r h_1$ and $x = s h_2$, some $h_1, h_2 \\in H$.\n\nThat is, $r h_1 = s h_2$, so $s^{-1} r h_1 = h_2$.\nSo $s^{-1} r = h_2 h_1^{-1}$, so $s^{-1} r$ is in $H$ by closure of $H$.\n\nBy taking inverses, $r^{-1} s$ is in $H$.\n\nBut that means $\\{ s h : h \\in H \\}$ and $\\{ r h : h \\in H\\}$ are equal.\nIndeed, we show that each is contained in the other.\n\n- if $a$ is in the right-hand side, then $a = rh$ for some $h$. Then $s^{-1} a = s^{-1} r h$; but $s^{-1} r$ is in $H$, so $s^{-1} r h$ is in $H$, and so $s^{-1} a$ is in $H$.\nTherefore $a \\in s H$, so $a$ is in the left-hand side.\n- if $a$ is in the left-hand side, then $a = sh$ for some $h$. Then $r^{-1} a = r^{-1} s h$; but $r^{-1} s$ is in $H$, so $r^{-1} s h$ is in $H$, and so $r^{-1} a$ is in $H$.\nTherefore $a \\in rH$, so $a$ is in the right-hand side.\n%%\n\nWe have shown that the "[4j4 cosets]" $gH$ are all completely disjoint and are all the same size, and that every element lies in a bucket; this completes the proof.',
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',
'EricBruylant'
],
childIds: [],
parentIds: [
'lagrange_theorem_on_subgroup_size'
],
commentIds: [],
questionIds: [],
tagIds: [],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [
{
id: '4507',
parentId: 'group_mathematics',
childId: 'lagrange_theorem_on_subgroup_size_intuitive',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-28 08:59:26',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4508',
parentId: 'group_order',
childId: 'lagrange_theorem_on_subgroup_size_intuitive',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-28 08:59:34',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
subjects: [
{
id: '4509',
parentId: 'left_cosets_partition_parent_group',
childId: 'lagrange_theorem_on_subgroup_size_intuitive',
type: 'subject',
creatorId: 'PatrickStevens',
createdAt: '2016-06-28 09:30:01',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4510',
parentId: 'left_cosets_biject',
childId: 'lagrange_theorem_on_subgroup_size_intuitive',
type: 'subject',
creatorId: 'PatrickStevens',
createdAt: '2016-06-28 09:30:17',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4511',
parentId: 'lagrange_theorem_on_subgroup_size_intuitive',
childId: 'lagrange_theorem_on_subgroup_size_intuitive',
type: 'subject',
creatorId: 'PatrickStevens',
createdAt: '2016-06-28 09:30:21',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
lenses: [],
lensParentId: 'lagrange_theorem_on_subgroup_size',
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: '19293',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'EricBruylant',
edit: '9',
type: 'newEdit',
createdAt: '2016-08-27 14:11:30',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: 'updated link to cardinality'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '18589',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'deleteTag',
createdAt: '2016-08-08 13:33:15',
auxPageId: 'needs_clickbait_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '18587',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '8',
type: 'newEdit',
createdAt: '2016-08-08 13:33:04',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '18586',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '7',
type: 'newEdit',
createdAt: '2016-08-08 13:32:24',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '17122',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'EricBruylant',
edit: '0',
type: 'newTag',
createdAt: '2016-07-19 02:03:58',
auxPageId: 'needs_clickbait_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14681',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '6',
type: 'newEdit',
createdAt: '2016-06-28 09:34:05',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14680',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '5',
type: 'newEdit',
createdAt: '2016-06-28 09:32:01',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14678',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'newTeacher',
createdAt: '2016-06-28 09:30:21',
auxPageId: 'lagrange_theorem_on_subgroup_size_intuitive',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14679',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'newSubject',
createdAt: '2016-06-28 09:30:21',
auxPageId: 'lagrange_theorem_on_subgroup_size_intuitive',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14677',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'newSubject',
createdAt: '2016-06-28 09:30:18',
auxPageId: 'left_cosets_biject',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14675',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'newSubject',
createdAt: '2016-06-28 09:30:02',
auxPageId: 'left_cosets_partition_parent_group',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14672',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '4',
type: 'newEdit',
createdAt: '2016-06-28 09:06:55',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14671',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '3',
type: 'newEdit',
createdAt: '2016-06-28 09:04:12',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14670',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '2',
type: 'newEdit',
createdAt: '2016-06-28 09:00:58',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14667',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'newParent',
createdAt: '2016-06-28 09:00:03',
auxPageId: 'lagrange_theorem_on_subgroup_size',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14668',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-28 09:00:03',
auxPageId: 'group_mathematics',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14669',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-28 09:00:03',
auxPageId: 'group_order',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14665',
pageId: 'lagrange_theorem_on_subgroup_size_intuitive',
userId: 'PatrickStevens',
edit: '1',
type: 'newEdit',
createdAt: '2016-06-28 09:00:01',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}