{ localUrl: '../page/alternating_group_five_is_simple.html', arbitalUrl: 'https://arbital.com/p/alternating_group_five_is_simple', rawJsonUrl: '../raw/4jf.json', likeableId: '2746', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'alternating_group_five_is_simple', edit: '8', editSummary: '', prevEdit: '7', currentEdit: '8', wasPublished: 'true', type: 'wiki', title: 'The alternating group on five elements is simple', clickbait: 'The smallest (nontrivial) simple group is the alternating group on five elements.', textLength: '1420', alias: 'alternating_group_five_is_simple', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-28 08:23:28', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-17 18:32:38', 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: '111', text: 'The [-4hf] $A_5$ on five elements is [4jc simple].\n\n# Proof\n\nRecall that $A_5$ has [3gg order] $60$, so [4jn Lagrange's theorem] states that any subgroup of $A_5$ has order dividing $60$.\n\nSuppose $H$ is a normal subgroup of $A_5$, which is not the trivial subgroup $\\{ e \\}$.\nIf $H$ has order divisible by $3$, then by [4l6 Cauchy's theorem] there is a $3$-[49f cycle] in $H$ (because the $3$-cycles are the only elements with order $3$ in $A_5$).\nBecause $H$ [4jw is a union of conjugacy classes], and because the $3$-cycles [4kv form a conjugacy class in $A_n$ for $n > 4$], $H$ would therefore contain *every* $3$-cycle; but then [4hs it would be the entire alternating group].\n\nIf instead $H$ has order divisible by $2$, then there is a double transposition such as $(12)(34)$ in $H$, since these are the only elements of order $2$ in $A_5$.\nBut then $H$ contains the entire conjugacy class so it contains every double transposition; in particular, it contains $(12)(34)$ and $(15)(34)$, so it contains $(15)(34)(12)(34) = (125)$.\nHence as before $H$ contains every $3$-cycle so is the entire alternating group.\n\nSo $H$ must have order exactly $5$, by [4jn Lagrange's theorem]; so it contains an element of order $5$ since [4jh prime order groups are cyclic].\n\nThe only such elements of $A_n$ are $5$-cycles; but the conjugacy class of a $5$-cycle is of size $12$, which is too big to fit in $H$ which has size $5$.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', 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