In this paper,we introduce the set of maximal subgroups with non-trivial core and their corresponding second maximal subgroups.A correlation characterization of the group class J_(pr)is presented by establishing the r...In this paper,we introduce the set of maximal subgroups with non-trivial core and their corresponding second maximal subgroups.A correlation characterization of the group class J_(pr)is presented by establishing the relationship between the core and the second maximal subgroups in these classifications.展开更多
A new approach to knowledge acquisition in incomplete information system with fuzzy decisions is proposed. In such incomplete information system, the universe of discourse is classified by the maximal tolerance classe...A new approach to knowledge acquisition in incomplete information system with fuzzy decisions is proposed. In such incomplete information system, the universe of discourse is classified by the maximal tolerance classes, and fuzzy approximations are defined based on them. Three types of relative reducts of maximal tolerance classes are then proposed, and three types of fuzzy decision rules based on the proposed attribute description are defined. The judgment theorems and approximation discernibility functions with respect to them are presented to compute the relative reduct by using Boolean reasoning techniques, from which we can derive optimal fuzzy decision rules from the systems. At last, three types of relative reducts of the system and their computing methods are given.展开更多
In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we ob...In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we obtain the correspoding convergent rosults.展开更多
A subgroup A of a p-group G is said to be soft in G if CG(A)=A and|NG(A)/A|=p.In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft;see Theorem A and Proposition 2.4.
Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and ...Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.展开更多
economy and qualitative game in which the constraint or preference correspondences are Q θ-majorized are obtained in locally convex topological vector spaces.
An existence theorem of maximal elements for a new type of preference correspondences which are Q(0)-majorized is given. Then some existence theorems of equilibrium for abstract economy and qualitative game in which t...An existence theorem of maximal elements for a new type of preference correspondences which are Q(0)-majorized is given. Then some existence theorems of equilibrium for abstract economy and qualitative game in which the constraint or preference correspondences are Q(0)-majorized are obtained in locally convex topological vector spaces.展开更多
The Wielandt subgroup of a group G, denoted by w（G）, is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either wi（G） = ζ...The Wielandt subgroup of a group G, denoted by w（G）, is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either wi（G） = ζi（G） for all integer i or wi（G） = ζi＋1（G） for every integer i, and w（G/K） = ζ（G/K） for every normal subgroup g in G with K ≠ 1. Meanwhile, a necessary and sufficient condition for a regular p-group of maximal class satisfying w（G） = ζ2（G） is given. Finally, the authors prove that the power automorphism group PAut（G） is an elementary abelian p-group if G is a non-abelian p- group with elementary ζ（G） ∩ζ1（G）.展开更多
This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from...This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces LP（w） to weighted Morrey spaces Mpq（ω） for 1 〈 q 〈 p 〈 ∞. As a corollary, if M is （weak） bounded on Mpq（ω）, then ω∈Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tε on weighted Morrey spaces. Finally, we show that ω∈Ap if and only if ω∈BMOp＇ （ω） for 1 ≤ p 〈 ∞ and 1/p ＋ 1/p＇ = 1.展开更多
Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the seco...Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).展开更多
Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,thi...Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.展开更多
Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are...Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.展开更多
S. Adnan has proved the fallowing theorem: Let G be a finite group. If G has exactly 2 conjugacy classes of maximal subgroups, then G is solvable. We extend the conjugacy classes to isomorphic classes, and further gen...S. Adnan has proved the fallowing theorem: Let G be a finite group. If G has exactly 2 conjugacy classes of maximal subgroups, then G is solvable. We extend the conjugacy classes to isomorphic classes, and further generalize them to same order classes of maximal subgroups. We obtain the following results.展开更多
The Bogomolov multiplier B0 （G） of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality o...The Bogomolov multiplier B0 （G） of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether＇s problem. We show that if G is a central product of G1 and G2, regarding Ki ≤ Z（Gi）,i = 1,2, and θ ： G1 →G2 is a group homomorphism such that its restriction θ｜K1 ： K1 → K2 is an isomorphism, then the triviality of Bo（G1/K1）, Bo（G1） and B0（G2） implies the triviality of Bo（G）. We give a positive answer to Noether＇s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 （with the usual requirement for the roots of unity）.展开更多
A group H is said to be autocapable if there exists a group M such that H is isomorphic to the absolute central factor group M/L(M)of M.In this paper,we first prove that if N is a characteristic subgroup of an autocap...A group H is said to be autocapable if there exists a group M such that H is isomorphic to the absolute central factor group M/L(M)of M.In this paper,we first prove that if N is a characteristic subgroup of an autocapable group H,then N is neither the generalized quaternion group nor the semi-dihedral group.Next,we give the classification of finite groups G if G/L(G)is a 2-group of maximal class.展开更多
For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.Thi...For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1237101812201236)+1 种基金the Fundamental Research Funds for the Central Universities(Grant No.B240201093/2013)the Natural Science Foundation of the Anhui Higher Education Institutions(Grant No.2022AH051907)。
文摘In this paper,we introduce the set of maximal subgroups with non-trivial core and their corresponding second maximal subgroups.A correlation characterization of the group class J_(pr)is presented by establishing the relationship between the core and the second maximal subgroups in these classifications.
基金supported by the National Natural Science Foundation of China (61070241)the Natural Science Foundation of Shandong Province (ZR2010FM035)Science Research Foundation of University of Jinan (XKY0808)
文摘A new approach to knowledge acquisition in incomplete information system with fuzzy decisions is proposed. In such incomplete information system, the universe of discourse is classified by the maximal tolerance classes, and fuzzy approximations are defined based on them. Three types of relative reducts of maximal tolerance classes are then proposed, and three types of fuzzy decision rules based on the proposed attribute description are defined. The judgment theorems and approximation discernibility functions with respect to them are presented to compute the relative reduct by using Boolean reasoning techniques, from which we can derive optimal fuzzy decision rules from the systems. At last, three types of relative reducts of the system and their computing methods are given.
文摘In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we obtain the correspoding convergent rosults.
基金supported by National Natural Science Foundation of China(Grant No.11071150)Natural Science Foundation of Shanxi Province(Grant No.2008012001)The Returned Abroad-student Found of Shanxi Province(Grant No.[2007]13–56)
文摘A subgroup A of a p-group G is said to be soft in G if CG(A)=A and|NG(A)/A|=p.In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft;see Theorem A and Proposition 2.4.
基金Supported by the National Natural Science Foundation of China(11871436,12071437)。
文摘Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.
文摘economy and qualitative game in which the constraint or preference correspondences are Q θ-majorized are obtained in locally convex topological vector spaces.
文摘An existence theorem of maximal elements for a new type of preference correspondences which are Q(0)-majorized is given. Then some existence theorems of equilibrium for abstract economy and qualitative game in which the constraint or preference correspondences are Q(0)-majorized are obtained in locally convex topological vector spaces.
基金supported by the National Natural Science Foundation of China (No. 11071155)the Key Disciplines of Shanghai Municipality (No. S30104)
文摘The Wielandt subgroup of a group G, denoted by w（G）, is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either wi（G） = ζi（G） for all integer i or wi（G） = ζi＋1（G） for every integer i, and w（G/K） = ζ（G/K） for every normal subgroup g in G with K ≠ 1. Meanwhile, a necessary and sufficient condition for a regular p-group of maximal class satisfying w（G） = ζ2（G） is given. Finally, the authors prove that the power automorphism group PAut（G） is an elementary abelian p-group if G is a non-abelian p- group with elementary ζ（G） ∩ζ1（G）.
基金supported by National Natural Science Foundation of China(Grant No.11661075)
文摘This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces LP（w） to weighted Morrey spaces Mpq（ω） for 1 〈 q 〈 p 〈 ∞. As a corollary, if M is （weak） bounded on Mpq（ω）, then ω∈Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tε on weighted Morrey spaces. Finally, we show that ω∈Ap if and only if ω∈BMOp＇ （ω） for 1 ≤ p 〈 ∞ and 1/p ＋ 1/p＇ = 1.
基金supported by the National Natural Science Foundation of China(Nos.11371232,11101252)the Shanxi Provincial Natural Science Foundation of China(No.2013011001)the Fundamental Research Funds for the Central Universities(No.BUPT2013RC0901)
文摘The groups as mentioned in the title are classified up to isomorphism. This is an answer to a question proposed by Berkovich and Janko.
文摘Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).
文摘Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.
文摘Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.
文摘S. Adnan has proved the fallowing theorem: Let G be a finite group. If G has exactly 2 conjugacy classes of maximal subgroups, then G is solvable. We extend the conjugacy classes to isomorphic classes, and further generalize them to same order classes of maximal subgroups. We obtain the following results.
基金Supported by Grant No.RD-08-82/03.02.2016 of Shumen University
文摘The Bogomolov multiplier B0 （G） of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether＇s problem. We show that if G is a central product of G1 and G2, regarding Ki ≤ Z（Gi）,i = 1,2, and θ ： G1 →G2 is a group homomorphism such that its restriction θ｜K1 ： K1 → K2 is an isomorphism, then the triviality of Bo（G1/K1）, Bo（G1） and B0（G2） implies the triviality of Bo（G）. We give a positive answer to Noether＇s problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 （with the usual requirement for the roots of unity）.
基金National Natural Science Foundation of China(12171302,11801334)Natural Science Foundation of Shanxi Province(202103021224287)+1 种基金Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi(2021L278)Science&Technology Development Fund of Tianjin Education Commission for Higher Education(2019KJ141).
文摘A group H is said to be autocapable if there exists a group M such that H is isomorphic to the absolute central factor group M/L(M)of M.In this paper,we first prove that if N is a characteristic subgroup of an autocapable group H,then N is neither the generalized quaternion group nor the semi-dihedral group.Next,we give the classification of finite groups G if G/L(G)is a 2-group of maximal class.
基金supported by National Natural Science Foundation of China(Grant Nos.11771258 and 11471198)。
文摘For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.
