{ localUrl: '../page/sign_of_permutation_is_well_defined.html', arbitalUrl: 'https://arbital.com/p/sign_of_permutation_is_well_defined', rawJsonUrl: '../raw/4hh.json', likeableId: '2736', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'sign_of_permutation_is_well_defined', edit: '3', editSummary: '', prevEdit: '2', currentEdit: '3', wasPublished: 'true', type: 'wiki', title: 'The sign of a permutation is well-defined', clickbait: 'This result is what allows the alternating group to exist.', textLength: '2315', alias: 'sign_of_permutation_is_well_defined', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-28 14:14:29', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-17 13:42:02', 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: '44', text: 'The [-497] $S_n$ contains elements which are made up from [4cn transpositions] ([4cp proof]).\nIt is a fact that if $\\sigma \\in S_n$ can be made by multiplying together an even number of transpositions, then it cannot be made by multiplying an odd number of transpositions, and vice versa.\n\n%%%knows-requisite([47y]):\nEquivalently, there is a [-47t] from $S_n$ to the [-47y] $C_2 = \\{0,1\\}$, taking the value $0$ on those permutations which are made from an even number of permutations and $1$ on those which are made from an odd number.\n%%%\n\n# Proof\n\nLet $c(\\sigma)$ be the number of cycles in the [49f disjoint cycle decomposition] of $\\sigma \\in S_n$, including singletons.\nFor example, $c$ applied to the identity yields $n$, while $c((12)) = n-1$ because $(12)$ is shorthand for $(12)(3)(4)\\dots(n-1)(n)$.\nWe claim that multiplying $\\sigma$ by a transposition either increases $c(\\sigma)$ by $1$, or decreases it by $1$.\n\nIndeed, let $\\tau = (kl)$.\nEither $k, l$ lie in the same cycle in $\\sigma$, or they lie in different ones.\n\n- If they lie in the same cycle, then $$\\sigma = \\alpha (k a_1 a_2 \\dots a_r l a_s \\dots a_t) \\beta$$ where $\\alpha, \\beta$ are disjoint from the central cycle (and [49g hence commute] with $(kl)$).\nThen $\\sigma (kl) = \\alpha (k a_s \\dots a_t)(l a_1 \\dots a_r) \\beta$, so we have broken one cycle into two.\n- If they lie in different cycles, then $$\\sigma = \\alpha (k a_1 a_2 \\dots a_r)(l b_1 \\dots b_s) \\beta$$ where again $\\alpha, \\beta$ are disjoint from $(kl)$.\nThen $\\sigma (kl) = \\alpha (k b_1 b_2 \\dots b_s l a_1 \\dots a_r) \\beta$, so we have joined two cycles into one.\n\nTherefore $c$ takes even values if there are evenly many transpositions in $\\sigma$, and odd values if there are odd-many transpositions in $\\sigma$.\n\nMore formally, let $\\sigma = \\alpha_1 \\dots \\alpha_a = \\beta_1 \\dots \\beta_b$, where $\\alpha_i, \\beta_j$ are transpositions.\n%%%knows-requisite([modular_arithmetic]):\n(The following paragraph is more succinctly expressed as: "$c(\\sigma) \\equiv n+a \\pmod{2}$ and also $\\equiv n+b \\pmod{2}$, so $a \\equiv b \\pmod{2}$.")\n%%%\nThen $c(\\sigma)$ flips odd-to-even or even-to-odd for each integer $1, 2, \\dots, a$; it also flips odd-to-even or even-to-odd for each integer $1, 2, \\dots, b$.\nTherefore $a$ and $b$ must be of the same [even_odd_parity parity].\n', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', 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