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Topic: Even and odd permutations

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	Encyclopedia :: encyclopedia : Even and odd permutations (Site not responding. Last check: 2007-09-10)
		An even permutation is one that can be produced by an even number of exchanges of two elements (these exchanges are called transpositions).
		the composition of an odd and an even permutation is odd
		Every permutation of odd order must be even; the converse is not true in general.
	www.hallencyclopedia.com /topic/Even_and_odd_permutations.html (914 words)

	Reference.com/Encyclopedia/Even and odd numbers
		The even numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only.
		An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.
		The parity of a permutation (as defined in abstract algebra) is the parity (even or odd) of the number of transpositions into which the permutation can be decomposed.
	www.reference.com /browse/wiki/Even_and_odd_numbers (879 words)

	Even and odd numbers - Wikipedia, the free encyclopedia
		In mathematics, the parity of an object refers to whether it is even or odd.
		An odd function, such as an odd power of a variable, gives the negative of its result when given the negative of an argument.
		It is possible for a function to be neither odd nor even, and for the case f(x) = 0, to be both odd and even.
	en.wikipedia.org /wiki/Even_and_odd_numbers (333 words)

	Even and Odd Permutations (Site not responding. Last check: 2007-09-10)
		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 (446 words)

	Permutations
		This is because (a) permutations are closed under the operation followed by, (b) since the operation is "followed by" it is associative, (c) the "leave everything where it is" permutation is the identity and (c) every permutation can be undone by "running the film backwards" thus all permutations have inverses.
		And the permutation corresponding to the product of two elements of G is the composition under followed by of the permutations corresponding to the two elements singly.
		If you were given a permutation of the eggs and told to rearrange the eggs into that permutation you might begin with the first place in the egg carton, find the numbered egg that went into that place and traded the egg in the first place with the egg that belonged there.
	members.tripod.com /~dogschool/permutation.html (2315 words)

	Even and odd numbers (Site not responding. Last check: 2007-09-10)
		A number expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd.
		In an odd base, the number is even or odd according to the sum of its digits.
		Parity is evenness or oddness of an integer.
	even-and-odd-numbers.kiwiki.homeip.net (635 words)

	even odd permutations
		An even permutation is one that requires and even number of "swaps", an odd permutation is one that requires an odd number of permutations.
		It is a fact that a permutation is either odd or even, meaning that if there is a path of even length from A to B, then all paths from A to B are of even length.
		We know that if B is an odd permutation of A then it can't be even as well, meaning that if there is a path of odd length from A to B, then all paths from A to B are of odd length.
	www.physicsforums.com /showthread.php?p=818660 (1566 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)

	Reference.com/Encyclopedia/Odd
		a permutation of a finite set is odd if it can be written as the composition of an odd number of transpositions; see even and odd permutations
		In probability theory, the odds in favor of an event is the ratio of the event's probability to the complementary probability.
		Odd, a name of Norse origin, the 10th most common male name in Norway (curiously, Even is also a common Norwegian name).
	www.reference.com /browse/wiki/Odd (220 words)

	43 QUINTILLION COMBINATIONS
		After an odd number of moves, the side cube is rotated through an odd permutation, whereas an even number of moves carries the side cube into an even permutation.
		Since a side cube cannot exhibit an odd and an even permutation simultaneously, there cannot be two paths back to start having odd and even lengths.
		If the corners are arranged in an odd permutation, then the sides will present an odd permutation (as a result of arranging the corners), and only odd permutations are accessible thereafter.
	www.eklhad.net /rubik/enumerat.html (1826 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)

	Elementary Row Operations
		(e) is -1 for an odd permutation, and 1 for an even permutation.
		One permutation describes the term from det(c), and the other is the magic numbers we use to find a subterm within the expansion of this term.
		This permutation is joined to the term of det(b), and the result is the permutation that arranges rows and columns in det(c).
	www.mathreference.com /la-det,erop.html (1198 words)

	php-deluxe.net - description Even and odd permutations
		Every permutation of odd order (group theory) must be even; the converse is not true in general.
		If σ is a given permutation, we define an inversion pair for σ to be a pair of indices (i, j) such that i σ(j).
		Now if we compose σ with the transposition (i, i +1) of two adjacent numbers, then, compared to σ, the new permutation σ(i, i +1) will have exactly one inversion pair less (in case (i, i +1) was an inversion pair for σ) or more (in case (i, i +1) was not an inversion pair).
	www.php-deluxe.net /wiwimod,index.page,Even-and-odd-permutations.htm (759 words)

	Permutation Groups
		Thus we have found that every even permutation corresponds to exactly one odd permutation under this "transformation" of appending (1, 2) to the beginning and every odd permutation is corresponded to by exactly one even permutation by this method.
		So once we have expressed a permutation as the product of disjoint cycles we can determine whether it is an even or odd permutation by noticing the number of cycles with an even number of elements.
		Since they contribute an odd number of transpositions to the permutation the total parity of transpositions, even or odd, will depend on the number of permutations with an even number of elements.
	members.tripod.com /~dogschool/permgroups.html (2347 words)

	Even and Odd Permutations (Site not responding. Last check: 2007-09-10)
		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)

	PlanetMath: examples of finite simple groups (Site not responding. Last check: 2007-09-10)
		It is not difficult to see (say, by Cauchy's theorem) that these are the only abelian simple groups.
		Because every permutation is either even or odd, and there is a bijection between the two (multiply every even permutation by a transposition), the index of
		This is version 12 of examples of finite simple groups, born on 2002-11-04, modified 2004-11-17.
	planetmath.org /encyclopedia/AlternatingGroup.html (354 words)

	Even and odd permutations biography .ms (Site not responding. Last check: 2007-09-10)
		The signature of a permutation is defined to be +1 if the permutation is even and -1 if it is odd.
		The identity permutation is an even permutation since it can be written as (1 2)(1 2).
		Now if we compose σ with the transposition (i, i+1) of two adjacent numbers, then, compared to σ, the new permutation σ(i, i+1) will have exactly one inversion pair less (in case (i,i+1) was an inversion pair for σ) or more (in case (i, i+1) was not an inversion pair).
	www.biography.ms /Even_permutation.html (858 words)

	Even and odd permutations - Wikipedia, the free encyclopedia
		The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of A
		Every permutation can be produced by a sequence of transpositions: with the first transposition we put the first element of the permutation in its proper place, the second transposition puts the second element right etc. Given a permutation σ, we can write it as a product of transpositions in many different ways.
		T1 is the transposition (i, i+1) of two adjacent numbers, so, compared to σ, the new permutation σ(i, i+1) will have exactly one inversion pair less (in case (i,i+1) was an inversion pair for σ) or more (in case (i, i+1) was not an inversion pair).
	en.wikipedia.org /wiki/Even_and_odd_permutations (1058 words)

	Permutations and combinations - Topics in precalculus
		For example, if twelve different things are permuted, then the number of their permutations is 479,001,600.
		Thus the number of permutations of 4 different things taken 4 at a time is 4!.
		This is the number of permutations of 10 different things taken 4 at a time.
	www.themathpage.com /aPreCalc/permutations-combinations.htm (1073 words)

	Even and odd permutations - Evenandoddpermutations
		Using the notation explained in the permutation article, we can write There are (infinitely) many other ways of writing σ as a composition of transpositions, for instance
		The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of An (in Sn).
		It is therefore unambiguous to call the elements of Sn represented by even-length words "even", and the elements represented by odd-length words "odd".
	www.kopete.org /Even-and-odd-permutations.html (914 words)

	even - definition of even by the Free Online Dictionary, Thesaurus and Encyclopedia. (via CobWeb/3.1 ... (Site not responding. Last check: 2007-09-10)
		even - equal in degree or extent or amount; or equally matched or balanced; "even amounts of butter and sugar"; "on even terms"; "it was a fifty-fifty (or even) split"; "had a fifty-fifty (or even) chance"; "an even fight"
		uneven - not even or uniform as e.g.
		This was a surprise even to the actors, and when they saw the table, they looked at one another in rapturous amazement.
	www.thefreedictionary.com.cob-web.org:8888 /even (969 words)

	Even and odd permutations (Site not responding. Last check: 2007-09-10)
		[The reason: if σ is even, then (12)σ is odd; if σ is odd, then (12)σ is even; the two maps are inverse to each other.] A cycle is even if and only if its length is odd.
		Every permutation can be produced by a sequence of transpositions: with the first transposition we put the first element of the permutation in its proper place, the second transposition puts the second element right etc. Every transposition can be written as a product of an odd number of transpositions of adjacent elements, e.g.
		Now if we compose σ with the transposition (''i'', i+1) of two adjacent numbers, then, compared to σ, the new permutation σ(''i'', i+1) will have exactly one inversion pair less (in case (''i'',''i''+1) was an inversion pair for σ) or more (in case (''i'', i+1) was not an inversion pair).
	even-and-odd-permutations.kiwiki.homeip.net (960 words)

	15 Puzzle - mathematical implications of Sam Lloyd's mind game
		Of course, the mathematicians know the answer: the parity of the permutation (rearrangement) which represents the actual position is preserved during the moves.
		Because it changes the cell's color, there is an even number of tiles between the initial and the final positions of the tile.
		Sam Lloyd knew that the 15-14 problem had no solution, because the start position was an odd permutation, while the end position was even (they differs by an inversion).
	www.holotronix.com /samlloyd15b.html (702 words)

	The On-Line Encyclopedia of Integer Sequences
		See A115198 for the parity of a(n,m) with 1 for even, 0 for odd (main entry).
		The parity of these numbers determines whether a conjugacy class of the symmetric group S_n, which is determined by its cycle structure, consists of even or odd permutations.
		The sequence of row lengths is A066898 (total number of even parts in all partitions of n.
	www.research.att.com /~njas/sequences/A115201 (210 words)

	Even
		a permutation of a finite set is called even if it can be written as a composition of an even number of transpositions; see even and odd permutations
		The high rates extend even to women born in the 1940s, challenging perceptions that people were more chaste in the past.
		Motorists are advised to stay off the roads, even though some are beginning to open, officials at the Weld County Emergency Operations Center said this morning.
	www.governpub.com /Languages-E/Even.php (736 words)

	Groups and Subgroups (Site not responding. Last check: 2007-09-10)
		Thus in the collection of permutations reachable from a solved cube, there are none that move, say, a corner to an edge.
		We will be interested in special subsets of groups that are themselves groups--in other words, a non-empty subset of the permutations so that any product of permutations in the subset is another permutation in the subset.
		If you are a beginner with Rubik's Cube and you want to practice with some operations that jumble the cube but do not jumble it into a nightmare, consider restricting yourself to a subgroup of all the allowable moves.
	mathcircle.berkeley.edu /BMC3/perm/node7.html (351 words)

	Square One The first steps...
		In the mean time read the lecture notes of W.D. Joyner, in particular about permutations and the definition of odd and even permutations.
		Since we know that the Cube-preserving motions are all even permutations, I had to look at the total group to solve this puzzle.
		Then we have obtained an even permutation of narrow parts and a single swap of two wide parts.
	web.inter.nl.net /users/C.Eggermont/Puzzels/SquareOne/index.html (789 words)

	40th IMO shortlist 1999/A3 solution
		It is easy to see that (2) is impossible for n = 3, because it requires an even permutation, whereas the result of 3 transpositions must be an odd permutation.
		We claim that half the permutations are such that the number of transpositions is always even and the other half are such that the number is always odd.
		Then any permutation either leaves D unchanged or changes D to -D. But a transposition changes D to -D, so an even number of transpositions leave D unchanged and an odd number change D to -D. That proves the claim except for the half.
	www.kalva.demon.co.uk /short/soln/sh99a3.html (799 words)

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