{
localUrl: '../page/cauchy_theorem_on_subgroup_existence.html',
arbitalUrl: 'https://arbital.com/p/cauchy_theorem_on_subgroup_existence',
rawJsonUrl: '../raw/4l6.json',
likeableId: '0',
likeableType: 'page',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
pageId: 'cauchy_theorem_on_subgroup_existence',
edit: '5',
editSummary: '',
prevEdit: '4',
currentEdit: '5',
wasPublished: 'true',
type: 'wiki',
title: 'Cauchy's theorem on subgroup existence',
clickbait: 'Cauchy's theorem is a useful condition for the existence of cyclic subgroups of finite groups.',
textLength: '2975',
alias: 'cauchy_theorem_on_subgroup_existence',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'PatrickStevens',
editCreatedAt: '2016-06-30 14:08:56',
pageCreatorId: 'PatrickStevens',
pageCreatedAt: '2016-06-18 15:36:35',
seeDomainId: '0',
editDomainId: 'AlexeiAndreev',
submitToDomainId: '0',
isAutosave: 'false',
isSnapshot: 'false',
isLiveEdit: 'true',
isMinorEdit: 'false',
indirectTeacher: 'false',
todoCount: '1',
isEditorComment: 'false',
isApprovedComment: 'true',
isResolved: 'false',
snapshotText: '',
anchorContext: '',
anchorText: '',
anchorOffset: '0',
mergedInto: '',
isDeleted: 'false',
viewCount: '89',
text: 'Cauchy's theorem states that if $G$ is a finite [-3gd] and $p$ is a [4mf prime] dividing $|G|$ the [3gg order] of $G$, then $G$ has a subgroup of order $p$. Such a subgroup is necessarily [47y cyclic] ([4jh proof]).\n\n# Proof\n\nThe proof involves basically a single magic idea: from thin air, we pluck the definition of the following set.\n\nLet $$X = \\{ (x_1, x_2, \\dots, x_p) : x_1 x_2 \\dots x_p = e \\}$$ the collection of $p$-[-tuple]s of elements of the group such that the group operation applied to the tuple yields the identity.\nObserve that $X$ is not empty, because it contains the tuple $(e, e, \\dots, e)$.\n\nNow, the cyclic group $C_p$ of order $p$ [3t9 acts] on $X$ as follows: $$(h, (x_1, \\dots, x_p)) \\mapsto (x_2, x_3, \\dots, x_p, x_1)$$ where $h$ is the generator of $C_p$.\nSo a general element $h^i$ acts on $X$ by sending $(x_1, \\dots, x_p)$ to $(x_{i+1}, x_{i+2} , \\dots, x_p, x_1, \\dots, x_i)$.\n\nThis is indeed a group action (exercise).\n\n%%hidden(Show solution):\n\n- It certainly outputs elements of $X$, because if $x_1 x_2 \\dots x_p = e$, then $$x_{i+1} x_{i+2} \\dots x_p x_1 \\dots x_i = (x_1 \\dots x_i)^{-1} (x_1 \\dots x_p) (x_1 \\dots x_i) = (x_1 \\dots x_i)^{-1} e (x_1 \\dots x_i) = e$$\n- The identity acts trivially on the set: since rotating a tuple round by $0$ places is the same as not permuting it at all.\n- $(h^i h^j)(x_1, x_2, \\dots, x_p) = h^i(h^j(x_1, x_2, \\dots, x_p))$ because the left-hand side has performed $h^{i+j}$ which rotates by $i+j$ places, while the right-hand side has rotated by first $j$ and then $i$ places and hence $i+j$ in total.\n%%\n\nNow, fix $\\bar{x} = (x_1, \\dots, x_p) \\in X$.\n\nBy the [4l8 Orbit-Stabiliser theorem], the [4v8 orbit] $\\mathrm{Orb}_{C_p}(\\bar{x})$ of $\\bar{x}$ divides $|C_p| = p$, so (since $p$ is prime) it is either $1$ or $p$ for every $\\bar{x} \\in X$.\n\nNow, what is the size of the set $X$?\n%%hidden(Show solution):\nIt is $|G|^{p-1}$.\n\nIndeed, a single $p$-tuple in $X$ is specified precisely by its first $p$ elements; then the final element is constrained to be $x_p = (x_1 \\dots x_{p-1})^{-1}$.\n%%\n\nAlso, the orbits of $C_p$ acting on $X$ partition $X$ ([4mg proof]).\nSince $p$ divides $|G|$, we must have $p$ dividing $|G|^{p-1} = |X|$.\nTherefore since $|\\mathrm{Orb}_{C_p}((e, e, \\dots, e))| = 1$, there must be at least $p-1$ other orbits of size $1$, because each orbit has size $p$ or $1$: if we had fewer than $p-1$ other orbits of size $1$, then there would be at least $1$ but strictly fewer than $p$ orbits of size $1$, and all the remaining orbits would have to be of size $p$, contradicting that $p \\mid |X|$.\n[todo: picture of class equation]\n\nHence there is indeed another orbit of size $1$; say it is the singleton $\\{ \\bar{x} \\}$ where $\\bar{x} = (x_1, \\dots, x_p)$.\n\nNow $C_p$ acts by cycling $\\bar{x}$ round, and we know that doing so does not change $\\bar{x}$, so it must be the case that all the $x_i$ are equal; hence $(x, x, \\dots, x) \\in X$ and so $x^p = e$ by definition of $X$.',
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: [
'cauchy_theorem_on_subgroup_existence_intuitive'
],
parentIds: [
'group_mathematics'
],
commentIds: [],
questionIds: [],
tagIds: [
'proof_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [
{
id: '4256',
parentId: 'cyclic_group',
childId: 'cauchy_theorem_on_subgroup_existence',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-18 15:12:56',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4258',
parentId: 'group_action',
childId: 'cauchy_theorem_on_subgroup_existence',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-18 15:23:02',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4279',
parentId: 'orbit_stabiliser_theorem',
childId: 'cauchy_theorem_on_subgroup_existence',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-19 17:29:27',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4302',
parentId: 'prime_number',
childId: 'cauchy_theorem_on_subgroup_existence',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-06-20 08:48:07',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
subjects: [],
lenses: [
{
id: '55',
pageId: 'cauchy_theorem_on_subgroup_existence',
lensId: 'cauchy_theorem_on_subgroup_existence_intuitive',
lensIndex: '0',
lensName: 'Intuitive version',
lensSubtitle: '',
createdBy: '267',
createdAt: '2016-06-30 15:52:00',
updatedBy: '267',
updatedAt: '2016-06-30 15:52:23'
}
],
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: '15159',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newTag',
createdAt: '2016-07-03 08:08:13',
auxPageId: 'proof_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14965',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newChild',
createdAt: '2016-06-30 15:51:58',
auxPageId: 'cauchy_theorem_on_subgroup_existence_intuitive',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14958',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '5',
type: 'newEdit',
createdAt: '2016-06-30 14:08:56',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14085',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '4',
type: 'newEdit',
createdAt: '2016-06-20 08:48:09',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14084',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-20 08:48:08',
auxPageId: 'prime_number',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14001',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '3',
type: 'newEdit',
createdAt: '2016-06-19 17:29:31',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '14000',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-19 17:29:27',
auxPageId: 'orbit_stabiliser_theorem',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13931',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequiredBy',
createdAt: '2016-06-18 15:39:28',
auxPageId: 'alternating_group_five_is_simple',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13930',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '2',
type: 'newEdit',
createdAt: '2016-06-18 15:36:47',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13927',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newParent',
createdAt: '2016-06-18 15:36:37',
auxPageId: 'group_mathematics',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13929',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-18 15:36:37',
auxPageId: 'group_action',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13925',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-06-18 15:36:36',
auxPageId: 'cyclic_group',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13923',
pageId: 'cauchy_theorem_on_subgroup_existence',
userId: 'PatrickStevens',
edit: '1',
type: 'newEdit',
createdAt: '2016-06-18 15:36:35',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'true',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}