Кіріспе
Топ теориясындағы қасиеттер
Property in group theory
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
Математикада, X ең болмағанда екі элементі бар шекті жиын болса, X-тің пермутациялары (яғни X-тен X-ке биективті функциялар) тең өлшемді екі классқа бөлінеді: жұп пермутациялар және тақ пермутациялар. Егер X-тің кез келген толық реті белгіленсе, пермутация σ-ның паритетін (жұп немесе тақ) σ үшін инверсиялар санының паритеті ретінде анықтауға болады, яғни x < y және σ(x) > σ(y шартын қанағаттандыратын X жиынының x, y элементтерінің жұптары үшін. Пермутация σ-ның белгісі, қолтаңбасы немесе сигнумы sgn(σ) деп белгіленеді және σ жұп болса +1, ал σ тақ болса −1 деп анықталады. Қолтаңба Sn симметриялық тобының алмасу белгісін анықтайды. Пермутация белгісін көрсетудің тағы бір тәсілі – жалпы Леви-Чивита символы (εσ) арқылы беріледі, ол X-тен X-ке дейінгі барлық бейнелеулер үшін анықталады және биективті емес бейнелеулер үшін нөлдік мәнге ие. Пермутация белгісі мына түрде нақты көрсетілуі мүмкін:
Property in group theory
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
sgn(σ) = (−1)^(N(σ))
Property in group theory
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
мұнда N(σ) – σ-дағы инверсиялар саны.
Property in group theory
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
Балама ретінде, пермутация σ-ның белгісін оны транспозициялардың көбейтіндісіне жіктеу арқылы анықтауға болады:
Property in group theory
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
sgn(σ) = (−1)^(m)
Property in group theory
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
мұнда m – жіктемедегі транспозициялар саны. Мұндай жіктеме бірегей болмаса да, барлық жіктемелердегі транспозициялар санының паритеті бірдей болады, бұл пермутация белгісі дұрыс анықталғанын көрсетеді.
Property in group theory
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
In mathematics, when X is a finite set with at least two elements, the permutations of X (i. e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation of X can be defined as the parity of the number of inversions for σ, i. e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non bijective maps. The sign of a permutation can be explicitly expressed as
1=sgn(σ) = (−1)^(N(σ))
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
1=sgn(σ) = (−1)^(m)
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined.
Қасиеттері
Тұлға алмасуы – жұп алмасу болып табылады. Сонымен қатар, жұп пермутациялар Sn-нің кіші тобын құрайтынын көреміз. Ол sgn гомоморфизмінің ядросы болып табылады. Тақ пермутациялардың қосындысы жұп болатындықтан, олар кіші топты құрай алмайды, бірақ An (Sn) косетін құрайды. Егер n > 1 болса, онда Sn-дегі жұп пермутациялардың саны тақ пермутациялардың санымен тең; соған сәйкес, An-да n!/2 пермутация бар. (Мұның себебі: егер σ жұп болса, онда (1 2)σ тақ, ал егер σ тақ болса, онда (1 2)σ жұп, және бұл екі түрлендіру бір-біріне кері.) Цикл оның ұзындығы тақ болса ғана жұп болады. Бұл сияқты формулалардан көрінеді: іс жүзінде, берілген пермутация жұп немесе тақ екенін анықтау үшін пермутацияны оқшау циклдардың көбейтіндісі түрінде жазады. Пермутация осы жіктеуде жұп ұзындықтағы циклдардың тақ саны болса ғана тақ болады. Берілген пермутация жұп немесе тақ екенін анықтаудың тағы бір тәсілі – сәйкес пермутациялық матрицаны құру және оның анықтамасын есептеу. Анықтаманың мәні пермутацияның тақтығымен бірдей. Тақ реті бар кез келген пермутация жұп болуы керек. А4-тегі (1 2)(3 4) пермутациясы керісіншесінің жалпы жағдайда дұрыс еместігін көрсетеді.
The identity permutation is an even permutation. Furthermore, we see that the even permutations form a subgroup of Sn. It is the kernel of the homomorphism sgn. 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). If n > 1, then there are just as many even permutations in Sn as there are odd ones; consequently, An contains n!/2 permutations. (The reason is that if σ is even then (1 2)σ is odd, and if σ is odd then (1 2)σ is even, and these two maps are inverse to each other.) A cycle is even if and only if its length is odd. This follows from formulas like
In practice, in order to determine whether a given permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation is odd if and only if this factorization contains an odd number of even length cycles. Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value of the determinant is the same as the parity of the permutation. Every permutation of odd order must be even. The permutation (1 2)(3 4) in A4 shows that the converse is not true in general.
In practice, in order to determine whether a given permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation is odd if and only if this factorization contains an odd number of even length cycles. Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value of the determinant is the same as the parity of the permutation. Every permutation of odd order must be even. The permutation (1 2)(3 4) in A4 shows that the converse is not true in general.
Жалпылау
Паритті Коксетер топтарына жалпылауға болады: генераторларды таңдауға байланысты (симметриялық топ үшін, жапсарлас транспозициялар) ұзындық функциясы ℓ(v) анықталады, содан кейін v → (−1)ℓ(v) функциясы жалпыланған таңба функциясын береді.
Parity can be generalized to Coxeter groups: one defines a length function ℓ(v), which depends on a choice of generators (for the symmetric group, adjacent transpositions), and then the function v ↦ (−1)ℓ(v) gives a generalized sign map.