{ localUrl: '../page/index_two_subgroup_of_group_is_normal.html', arbitalUrl: 'https://arbital.com/p/index_two_subgroup_of_group_is_normal', rawJsonUrl: '../raw/4hl.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'index_two_subgroup_of_group_is_normal', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Index two subgroup of group is normal', clickbait: 'An easy (though not very widely applicable) criterion for a subgroup to be normal.', textLength: '828', alias: 'index_two_subgroup_of_group_is_normal', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-17 14:47:34', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-17 14:47:34', seeDomainId: '0', editDomainId: '163', 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: '16', text: 'Let $H$ be a [-subgroup] of the [-3gd] $G$, of [index_of_subgroup index] $2$.\nThen $H$ is a [-4h6] of $G$.\n\n# Proof\n\nWe must show that $H$ is closed under [4gk conjugation] by elements of $G$.\n\nSince $H$ has index $2$ in $G$, there are two left [group_coset cosets]: $H$ and $xH$ for some specific $x$.\nThere are also two right cosets: $H$ and $Hy$.\n\nNow, since $x \\not \\in H$, it must be the case that $x \\in Hy$; so without loss of generality, $x = y$.\n\nHence $xH = Hx$ and so $xHx^{-1} = H$.\n\nIt remains to show that $H$ is closed under conjugation by *every* element of $G$.\nBut every element of $G$ is either in $H$, or in $xH$; so it is either $h$ or $xh$, for some $h \\in H$.\n\n- $hHh^{-1}$ is equal to $H$ since $hH = H$ and $Hh^{-1} = H$.\n- $xh H (xh)^{-1} = xhHh^{-1} x^{-1} = xHx^{-1} = H$.\n\nThis completes the proof.\n', 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' ], childIds: [], parentIds: [], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], 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: '13519', pageId: 'index_two_subgroup_of_group_is_normal', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-06-17 14:47:34', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'false', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }