{ localUrl: '../page/order_of_a_group_element.html', arbitalUrl: 'https://arbital.com/p/order_of_a_group_element', rawJsonUrl: '../raw/4cq.json', likeableId: '2699', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'EricRogstad' ], pageId: 'order_of_a_group_element', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Order of a group element', clickbait: '', textLength: '1097', alias: 'order_of_a_group_element', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-15 10:14:47', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-15 10:14:47', 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: '26', text: 'Given an element $g$ of group $(G, +)$ (which henceforth we abbreviate simply as $G$), the order of $g$ is the number of times we must add $g$ to itself to obtain the identity element $e$.\n\n%%%knows-requisite([3gg]):\nEquivalently, it is the order of the group $\\langle g \\rangle$ generated by $g$: that is, the order of $\\{ e, g, g^2, \\dots, g^{-1}, g^{-2}, \\dots \\}$ under the inherited group operation $+$.\n%%%\n\nConventionally, the identity element itself has order $1$.\n\n# Examples\n\n%%%knows-requisite([497]):\nIn the [-497] $S_5$, the order of an element is the [-least_common_multiple] of its [4cg cycle type].\n%%%\n%%%knows-requisite([47y]):\nIn the [-47y] $C_6$, the order of the generator is $6$.\nIf we view $C_6$ as being the integers [modular_arithmetic modulo] $6$ under addition, then the element $0$ has order $1$; the elements $1$ and $5$ have order $6$; the elements $2$ and $4$ have order $3$; and the element $3$ has order $2$.\n%%%\n\nIn the group $\\mathbb{Z}$ of [48l integers] under addition, every element except $0$ has infinite order. $0$ itself has order $1$, being the identity.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: [ '0', '0', '0', '0', '0', '0', '0', '0', '0', '0' ], 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' ], childIds: [], parentIds: [ 'group_mathematics' ], commentIds: [], questionIds: [], tagIds: [ 'formal_definition_meta_tag', 'needs_clickbait_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [ { id: '3975', parentId: 'order_of_a_group_element', childId: 'group_order', type: 'subject', creatorId: 'AlexeiAndreev', createdAt: '2016-06-17 21:58:56', level: '1', isStrong: 'false', everPublished: 'true' } ], 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: '17127', pageId: 'order_of_a_group_element', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-19 02:06:08', auxPageId: 'needs_clickbait_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13743', pageId: 'order_of_a_group_element', userId: 'EricBruylant', edit: '0', type: 'deleteTag', createdAt: '2016-06-17 22:02:18', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13741', pageId: 'order_of_a_group_element', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-06-17 22:02:17', auxPageId: 'formal_definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13006', pageId: 'order_of_a_group_element', userId: 'PatrickStevens', edit: '1', type: 'newTeacher', createdAt: '2016-06-15 10:15:44', auxPageId: 'group_order', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13005', pageId: 'order_of_a_group_element', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-06-15 10:14:47', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13002', pageId: 'order_of_a_group_element', userId: 'PatrickStevens', edit: '1', type: 'newTag', createdAt: '2016-06-15 10:07:58', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13001', pageId: 'order_of_a_group_element', userId: 'PatrickStevens', edit: '1', type: 'newParent', createdAt: '2016-06-15 10:07:50', auxPageId: 'group_mathematics', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }