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Topic: Even permutation

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Odd permutation

Permutation

Permutations and combinations

	Even and odd permutations - Wikipedia, the free encyclopedia
		An even permutation is one that can be produced by an even number of exchanges of two elements (these exchanges are called transpositions).
		The identity permutation is an even permutation since it can be written as (1 2)(1 2).
		Every permutation of odd order must be even; the converse is not true in general.
	en.wikipedia.org /wiki/Even_permutation (902 words)

	Permutation article - Permutation permutation (music) mathematics change ringing alphabet string - What-Means.com (Site not responding. Last check: 2007-11-05)
		A permutation of the alphabet of 26 letters is a string of length 26 containing each letter just once; and it is clear that this definition works for any alphabet of N letters, with strings of length N.
		That is, a permutation is simply a sequence with no two elements the same, drawn from a fixed set of symbols, and of maximum length.
		An even permutation is a permutation which can be expressed as the product of an even number of transpositions, and the identity permutation is an even permutation as it equals (1 2)(1 2).
	www.what-means.com /encyclopedia/Permutation (1562 words)

	Permutation - Wikipedia, the free encyclopedia
		In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself.
		A permutation is an ordered sequence containing each symbol from a set once and only once; neither (1, 2, 2, 3, 4, 5, 6) nor (1, 2, 4, 5, 6) are permutations.
		The "canonical" form for a permutation places the lowest-numbered position in each cycle first in that cycle and then orders the cycles by increasing first element; see also cycles and fixed points.
	en.wikipedia.org /wiki/Permutation (1389 words)

	[No title]
		permutation +------------------------------------------------------------ If S is a set, then a permutation of S is a bijective map from S into S. The intuition underlying the definition of a permutation is that a permutation determines a reordering of the elements in a list or the rearrangement of objects.
		Thus the identity permutation is an even permutation (since it is equal to the composition of any transposition with itself), and any transposition is an odd permutation (since it is equal to one transposition).
		That is, an even permutation is the composition of an even number of transpositions.
	www.math.harvard.edu /~knill/sofia/data/group.txt (4457 words)

	Even and odd numbers
		The number zero is even, because it is equal to two multiplied by zero.
		A number expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd.
		An integer is even if it is congruent to 0 modulo this ideal, in other words if it's congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.
	www.brainyencyclopedia.com /encyclopedia/e/ev/even_and_odd_numbers.html (469 words)

	Even and odd permutations -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		An even permutation is one that can be produced by an (Click link for more info and facts about even number) even number of exchanges of two elements (these exchanges are called ((music) playing in a different key from the key intended; moving the pitch of a piece of music upwards or downwards) transpositions).
		Furthermore, we see that the even permutations form a ((mathematics) a subset (that is not empty) of a mathematical group) subgroup of S
		Every permutation of odd ((biology) taxonomic group containing one or more families) order must be even; the converse is not true in general.
	www.absoluteastronomy.com /encyclopedia/e/ev/even_and_odd_permutations.htm (1028 words)

	Even and Odd Permutations (Site not responding. Last check: 2007-11-05)
		a transposition, the parity of the permutation is reversed.
		The first is achieved using an even number of transpositions, and so is the second, hence the composition is built using an even number of transpositions, and is even.
		The even permutations form the kernel of the parity homomorphism, and are a normal subgroup in g.
	www.mathreference.com /grp,eop.html (317 words)

	Rubik's Cube
		This has to follow since every move is an odd permutation and an odd permutation followed by an odd permutation is an even permutation.
		But any process with an odd numbers of moves cannot represent an even permutation of the edge cubelets since every move is a four-cycle (an odd permutation) of edge cubelets.
		From this we can conclude that (1) there is no process that can create an odd permutation of the edge facelets and that (2) an odd permutation of the corner facelets requires an odd number of moves and an even permutation of the corner facelets requires an even number of moves.
	members.tripod.com /~dogschool/rubikscube.html (2033 words)

	The Dynamic Transposition Combiner
		Permutation has been used for entire ciphers (mainly in an era of pencil-and-paper operations), and, in a limited form, is still in common use inside substitution-permutation networks [11] of the sort from which the U.S. Data Encryption Standard [e.g., 21] is built.
		Even though half of the exchange operations will have no effect, the number of effective bit-changes is still on the same order as the number of bits (two bits change on every effective exchange).
		Even an exact copy of the original plaintext block would seem to be of little help: There are a multitude of deciphering-but-incorrect permutations (too many to try them all), with apparently no way to identify the correct one.
	www.ciphersbyritter.com /ARTS/DYNTRAN2.HTM (6039 words)

	Determinants
		If this permutation is a sequence of an even number of transpositions, it is impossible to write this permutation as a sequence of an odd number of transpositions.
		If a permutation of n ordered elements can be expressed as an even number of transpositions, then it is called an even permutation.
		If a permutation of n ordered elements can be expressed as an odd number of transpositions, then it is called an odd permutation.
	www.ping.be /~ping1339/determ.htm (1885 words)

	Even and Odd Permutations
		The identity is an even permutation (0 2-cycles), and clearly if you multiply any even permutation by another even permutation, you will get an even permutation.
		Thus the set of all permutations that are even form a subset of the full symmetric group.
		This can be proved by showing that the sliding operation is like a permutation group, and that the swapping of two blocks amounts to an odd permutation in that group, but the operation of sliding a block is an even permutation.
	mathcircle.berkeley.edu /BMC3/perm/node8.html (319 words)

	Alternating groups.
		This means that all permutations could be revritten as alternations, and that all these altenations could be done against one element.
		A1 gives us that a circular permutation involving n elements will be possible to do in n-1 alternations.
		The identity element (1) will be an even permutation, because 0 is even.
	hemsidor.torget.se /users/m/mauritz/math/alg/alt.htm (274 words)

	[No title]
		A "permutation" of T is a bijection from T to itself.
		The first is the "matrix notation": To a permutation f : T --> T, given by [ 1 2...
		] [ n ] [ f(n) ] Furthermore, the inverse of the matrix of the permutation is the matrix of the inverse of the permutation: (b) P(f)^(-1) = P(f^(-1)).
	web.usna.navy.mil /~wdj/sm485_2.txt (1245 words)

	PlanetMath: simplicity of the alternating groups (Site not responding. Last check: 2007-11-05)
		We will extensively employ cycle notation, with composition on the left, as is usual.
		For the reverse inclusion, by definition every even permutation is the product of even number of
		This is version 9 of simplicity of the alternating groups, born on 2002-11-04, modified 2004-02-25.
	planetmath.org /encyclopedia/SimplicityOfA_n.html (316 words)

	Analysis of the Sixteen Puzzle
		THEOREM 1.1c: If n is even, then every legal configuration with the hole in row i where i is odd corresponds to a C[] permutation with an odd number of inversions.
		The equivalent permutation is 4, 5, 1, 3, 2.
		The equivalent permutation is 7, 2, 1, 4, 6, 3, 5.
	kevingong.com /Math/SixteenPuzzle.html (2870 words)

	New Page 2 (Site not responding. Last check: 2007-11-05)
		Let c be a fixed odd permutation, and consider any odd permutation a.
		The inverse permutation c' is also odd, and hence c'a = b, which is an even permutation.
		This implies that a = cb, where b is an even permutation.
	cas.memphis.edu /rfaudree/hom44261.htm (199 words)

	DeVill
		The blue one permutes the numbers in such a way, that you can solve that game, while the red one permutes so, that it is not possible to put the numbers in ascending order.
		So if we start with an odd permutation and the empty space in the first row, than every time when the empty space is in the first row the permutation will be odd.
		The game is solvable for every even permutation whose first number is 1, and the empty space is in the first row.
	www.cs.elte.hu /~devill/permjen.html (1620 words)

	Groups of Permutations
		Permutation is simply scrambling or reshuffling of a given set of items.
		Unscrambling of a permutation is itself a permutation.
		The Lemma is obvious since an even permutation is a product of an even number of transpositions.
	www.cut-the-knot.com /do_you_know/sgroups.shtml (523 words)

	3-D Crystals XI
		A permutation of a set of objects, placed in an initial sequence, means a replacement of one or more of those objects by objects from that same set, such that no two or more objects are replaced by the same object, a replacement resulting in a different sequence of those objects, i.e.
		Similarly an even permutation will decompose (in an unlimited variety of ways) into transpositions, the number of which is bound to be even.
		The odd permutations are not closed under successive application because (2p + 1) + (2q + 1) = 2p + 2q + 2 = 2(p + q + 1), which is even, and thus we are brought outside the set of odd permutations.
	home.hetnet.nl /~turing/d3_lattice_11.html (3432 words)

	Introduction
		Permutations that are not even are odd, and every representation of an odd permutation as a composition of transpositions contains an odd number of transpositions.
		Each term is associated with a distinct permutation, p, of set { 1, 2, , n }, where n is the number of nodes of G.
		In particular, the identity permutation provides one term, and because the identity permutation is even and leaves all operands fixed, we have from Result 3 that t(G, I) is either 0 or (—x)
	www.mathpuzzle.com /characteristic.html (2118 words)

	PlanetMath: signature of a permutation
		"signature of a permutation" is owned by rspuzio.
		Cross-references: induction, identity map, clear, relation, totally ordered, odd, transposition, multiplicative group, homomorphism, permutation group, permutations, group, finite
		This is version 6 of signature of a permutation, born on 2003-02-26, modified 2004-10-31.
	planetmath.org /encyclopedia/SignatureOfAPermutation.html (118 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		We wish to prove that the plus sign should be used in (9) if both of the permutations of (10) are even or both are odd, and that the minus sign should be used when one is even and the other odd.
		with the top one even, and hence the new 2-permutation rule indicates that the sign should be plus when the bottom permutation is even and minus otherwise.
		Whenever one such interchange is made, the row subscript permutation and column subscript permutation will each change from even to odd or from odd to even.
	www.thiel.edu /Mathproject/atps/chptr09/p089.htm (259 words)

	Symmetric group - SmartyBrain Encyclopedia and Dictionary
		A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition.
		The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd.
		The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd.
	smartybrain.com /index.php/Symmetric_group (631 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		In an article dated 5/13/01, Fred Galvin posted a short proof (quoted below) that every even permutation is a product of two n-cycles, which implies that every element of A_n is the commutator of an n-cycle with something else: aba^-1b^-1 = a(ba^-1b^-1).
		Every even permutation in S_n is the product of two n-cycles.
		[Composition of permutations from right to left.] Then g is an even permutation leaving n fixed.
	www.math.niu.edu /~rusin/known-math/01_incoming/alt_n_commut (200 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		A given permutation can be represented by % many different such transposition sequences, but the number of % such transpositions will always be odd or always be even.
		% If the number of transpositions is even or odd, the permutation is % said to be even or odd.
		% +1, the permutation is even, % -1, the permutation is odd.
	www.csit.fsu.edu /~burkardt/math2071/perm_sign.m (280 words)

	Order of a permutation (Site not responding. Last check: 2007-11-05)
		One important application of transpositions is that every permutation may be written as a product of transpositions (although not necessarily disjoint and not uniquely).
		As the definition suggests, a permutation cannot be both odd and even.
		Of course, the identity permutation is an even permutation.
	www.math.csusb.edu /notes/advanced/algebra/gp/node7.html (148 words)

	: Answers to problems2
		To prove that the order of An is 1/2(n!) it suffices to prove that Sn has the same number of even permutation as odd permutations, since Sn has order n!.
		as follow: take any even permutation a and define f(a) to be the composition (the product) a(12), where (12) is the two cycle sending 1 to 2 and 2 to 1.
		); that is, we would have that the number of even permutations is the same as the number of odd permutations.
	home.cc.umanitoba.ca /discus/messages/220/240.html?1037076368 (497 words)

	Useful Mathematics
		At first sight this is not a simple permutation group because the orientation of the pieces matters, but you could consider it a permutation group of the 48 moving facelets instead of the 20 moving pieces.
		Each quarter turn is an odd permutation on the corners of the cube (and also odd on the edges), so it is impossible for a face centre to move a quarter turn without any other pieces moving.
		The parity of the permutation of all 20 pieces (corners and edges) must be even for the cube position to be solvable.
	www.geocities.com /jaapsch/puzzles/theory.htm (7326 words)

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