Let G be a finite group with the property that for any conjugacy class order, G has exactly two conjugacy classes which have the same order. We prove that： （1） ff a Sylow 2-subgroup of G is Abelian, then G is isomo...Let G be a finite group with the property that for any conjugacy class order, G has exactly two conjugacy classes which have the same order. We prove that： （1） ff a Sylow 2-subgroup of G is Abelian, then G is isomorphic to the direct product of symmetric group with order 3 and cyclic group with order 2, or G is isomorphic to the semidirect product of a cyclic group with order 3 and a cyclic group with order 4; （2） if G＇ is nilpotent, then G is a group of {2,3,5 }.展开更多
In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties,...In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.展开更多
Let f ： U（x0） belong to E → F be a C^1 map and f＇（x0） be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surject...Let f ： U（x0） belong to E → F be a C^1 map and f＇（x0） be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties： f＇（x0） is double splitting and R（f＇（x）） ∩ N（T0^＋） = {0} near x0. However, in infinite dimensional Banach spaces, f＇（x0） is not always double splitting （i.e., the generalized inverse of f（x0） does not always exist）, but its bounded outer inverse of f＇（x0） always exists. Only using the C^1 map f and the outer inverse To^# of f（x0）, the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.展开更多
The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N wh...The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the Langlands functoriality.展开更多
Applications of locally fine property for operators are further developed. Let E and F be Banach spaces and f: U( x0) E—→F be C1 nonlinear map, where U (x0) is an open set containing point x0∈ E. With the locally f...Applications of locally fine property for operators are further developed. Let E and F be Banach spaces and f: U( x0) E—→F be C1 nonlinear map, where U (x0) is an open set containing point x0∈ E. With the locally fine property for Frechet derivatives f’ (x) and generalized rank theorem for f ’( x), a local conjugacy theorem, i. e. a characteristic condition for f being conjugate to f (x0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.展开更多
Let E and F be Banach spaces and f non-linear C1 map from E into F. The main result isTheorem 2.2, in which a connection between local conjugacy problem of f at x0E and a localfine property of f'(x) at x0(see the ...Let E and F be Banach spaces and f non-linear C1 map from E into F. The main result isTheorem 2.2, in which a connection between local conjugacy problem of f at x0E and a localfine property of f'(x) at x0(see the Definition 1.1 in this paper) are obtained. This theoremincludes as special cases the two known theorems: the finite rank theorem and Berger's Theoremfor non-linear Fredholm operators. Moreover, the thcorem gives rise the further results for somenon-linear semi-Fredholm maps and for all non-linear semi-Wedholm maps when E and F areHilbert spaces. Thus Theorem 2.2 not only just unifies the above known theorems but alsoreally generalizes them.展开更多
It was proved that all continuous functions are topologically conjugate to their continuous iterative roots in monotonic cases. An interesting problem reads: Does the same conclusion hold in non-monotonic cases?We giv...It was proved that all continuous functions are topologically conjugate to their continuous iterative roots in monotonic cases. An interesting problem reads: Does the same conclusion hold in non-monotonic cases?We give a negative answer to the problem by presenting a necessary condition for the topological conjugacy,which helps us construct counter examples. We also give a sufficient condition as well as a method of constructing the topological conjugacy.展开更多
Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R ...Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.展开更多
This note will give some classification results for local and global C^1 diffeomorphisms of R under C^1 conjugacy. For γ=1, 2,…, ∞, ω, let D~γ(0)={f: R→R is C~γ; f’(x)】0, x∈R; and f has 0 asits unique fixed ...This note will give some classification results for local and global C^1 diffeomorphisms of R under C^1 conjugacy. For γ=1, 2,…, ∞, ω, let D~γ(0)={f: R→R is C~γ; f’(x)】0, x∈R; and f has 0 asits unique fixed point} (local diffeomorphisms of R). In Refs. [1] and [2], we have considered the existence of smooth embedding flows and the other related problems for f∈D~γ(0). As a result, we prove that the following classifications have some numerical invariants展开更多
Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E_n(G)of all elements of order n in G is a conj...Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E_n(G)of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E_n(G) are also given.展开更多
In this paper,a topological classification of piecewise monotone functions(abbreviated as PM functions)with nonmonotonicity height equal to 1 which are strictly increasing on their characteristic intervals and have fi...In this paper,a topological classification of piecewise monotone functions(abbreviated as PM functions)with nonmonotonicity height equal to 1 which are strictly increasing on their characteristic intervals and have finitely many fixed points is presented.展开更多
Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We show that if 4 divides no A-invariant conjugacy class size of G, then G is solvable. We also characterize the A-invariant struc...Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We show that if 4 divides no A-invariant conjugacy class size of G, then G is solvable. We also characterize the A-invariant structure of G under certain arithmetical conditions on the set of A-invariant class sizes of G by means of the fixed point subgroup, some of which imply the solvability of G. Thus, we extend, for coprime action, several results appeared in the literature on class sizes.展开更多
Let G be a group,and let →/ΓC(G) be a digraph whose vertices are the nontrivial conjugacy classes of G and there is an arc from a vertex C to a vertex C' if and only if C≠C' and(C)■(C').In this paper,w...Let G be a group,and let →/ΓC(G) be a digraph whose vertices are the nontrivial conjugacy classes of G and there is an arc from a vertex C to a vertex C' if and only if C≠C' and(C)■(C').In this paper,we characterize finite groups G whose associated digraphs →/ΓC(G) are oriented trees.展开更多
An element x of a finite group G is said to be primary if the order of x is a prime power.We define csp2(G)as follows:if|x^(G)|is a prime power for every primary element x of G,where x^(G) is the conjugacy class of x ...An element x of a finite group G is said to be primary if the order of x is a prime power.We define csp2(G)as follows:if|x^(G)|is a prime power for every primary element x of G,where x^(G) is the conjugacy class of x in G,then csp2(G)=0;if there exists a primary element x in G such that|x^(G)|is divisible by at least two distinct primes,then csp2(G)=max{|x^(G)||x∈Gis primary,|x^(G)|is divisible by at least two distinct primes}.In this paper we discuss the influence of the number csp2(G)on the structure of G.展开更多
Let G be a finite group and N a normal subgroup of G.Denote by Γ_(G)(N)the graph whose vertices are all distinct G-conjugacy class sizes of non-central elements in N,and two vertices of Γ_(G)(N)are adjacent if and o...Let G be a finite group and N a normal subgroup of G.Denote by Γ_(G)(N)the graph whose vertices are all distinct G-conjugacy class sizes of non-central elements in N,and two vertices of Γ_(G)(N)are adjacent if and only if they are not coprime numbers.We prove that if the center Z(N)=Z(G)∩N and Γ_(G)(N)is k-regular for k≥1,then either a section of Nis a quasi-Frobenius group or Γ_(G)(N)is a complete graph with k+1 vertices.展开更多
In this paper,an adaptive three-term conjugate gradient method is proposed for solving unconstrained problems,which generates sufficient descent directions at each iteration.Different from the existent methods,a dynam...In this paper,an adaptive three-term conjugate gradient method is proposed for solving unconstrained problems,which generates sufficient descent directions at each iteration.Different from the existent methods,a dynamical adjustment between Hestenes–Stiefel and Dai–Liao conjugacy conditions in our proposed method is developed.Under mild condition,we show that the proposed method converges globally.Numerical experimentation with the new method indicates that it efficiently solves the test problems and therefore is promising.展开更多
Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for etale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same ...Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for etale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.展开更多
Nowadays some promising authenticated group key agreement protocols are constructed on braid groups, dynamic groups, pairings and bilinear pairings. Hence the non-abelian structure has attracted cryptographers to cons...Nowadays some promising authenticated group key agreement protocols are constructed on braid groups, dynamic groups, pairings and bilinear pairings. Hence the non-abelian structure has attracted cryptographers to construct public-key cryptographic protocols. In this article, we propose a new authenticated group key agreement protocol which works in non-abelian near-rings. We have proved that our protocol meets the security attributes under the assumption that the twist conjugacy search problem（TCSP） is hard in near-ring.展开更多
Let m, n 〉 1 be two coprime integers. In this paper, we prove that a finite solvable group is nilpotent if the set of the conjugacy class sizes of its primary and biprimary elements is {1, rn, n, mn}.
Let G be a finite group and π be a set of primes including at least two elements. We write cd（G） and cs（G） to denote the set of complex irreducible character degrees and conjugacy class sizes of G , respectively,...Let G be a finite group and π be a set of primes including at least two elements. We write cd（G） and cs（G） to denote the set of complex irreducible character degrees and conjugacy class sizes of G , respectively, and write π（m）to denote the set of all prime divisors of a positive integer m . For any 1≠m∈cd（G） and 1≠m∈cs（G）, in this note, we shall present the corresponding group structures of finite group G in the case π（m）=π , respectively, which generalizes the result of finite groups with character degrees of two distinct primes. Furthermore, we shall see that the influence of the two sets on the group structure is analogous.展开更多
基金The Natural Science Foundation ofChongqing Education Committee (No.KG051107)
文摘Let G be a finite group with the property that for any conjugacy class order, G has exactly two conjugacy classes which have the same order. We prove that： （1） ff a Sylow 2-subgroup of G is Abelian, then G is isomorphic to the direct product of symmetric group with order 3 and cyclic group with order 2, or G is isomorphic to the semidirect product of a cyclic group with order 3 and a cyclic group with order 4; （2） if G＇ is nilpotent, then G is a group of {2,3,5 }.
文摘In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.
基金Project supported by the National Natural Science Foundation of China (No. 10271053).
文摘Let f ： U（x0） belong to E → F be a C^1 map and f＇（x0） be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties： f＇（x0） is double splitting and R（f＇（x）） ∩ N（T0^＋） = {0} near x0. However, in infinite dimensional Banach spaces, f＇（x0） is not always double splitting （i.e., the generalized inverse of f（x0） does not always exist）, but its bounded outer inverse of f＇（x0） always exists. Only using the C^1 map f and the outer inverse To^# of f（x0）, the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.
基金National Natural Science Foundation of China (Grant No. A010102-11671380)One Hundred Talents Program at Chinese Academy of Sciences, National Basic Research Program of China (Grant No. 2013CB834202)National Science Foundation of USA (Grant No. DMS9729992)。
文摘The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the Langlands functoriality.
文摘Applications of locally fine property for operators are further developed. Let E and F be Banach spaces and f: U( x0) E—→F be C1 nonlinear map, where U (x0) is an open set containing point x0∈ E. With the locally fine property for Frechet derivatives f’ (x) and generalized rank theorem for f ’( x), a local conjugacy theorem, i. e. a characteristic condition for f being conjugate to f (x0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.
文摘Let E and F be Banach spaces and f non-linear C1 map from E into F. The main result isTheorem 2.2, in which a connection between local conjugacy problem of f at x0E and a localfine property of f'(x) at x0(see the Definition 1.1 in this paper) are obtained. This theoremincludes as special cases the two known theorems: the finite rank theorem and Berger's Theoremfor non-linear Fredholm operators. Moreover, the thcorem gives rise the further results for somenon-linear semi-Fredholm maps and for all non-linear semi-Wedholm maps when E and F areHilbert spaces. Thus Theorem 2.2 not only just unifies the above known theorems but alsoreally generalizes them.
基金supported by National Natural Science Foundation of China(Grant Nos.11301226 and 11301572)Zhejiang Provincial Natural Science Foundation of China(Grant No.LQ13A010017)Chongqing Normal University Project(Grant No.13XLZ04)
文摘It was proved that all continuous functions are topologically conjugate to their continuous iterative roots in monotonic cases. An interesting problem reads: Does the same conclusion hold in non-monotonic cases?We give a negative answer to the problem by presenting a necessary condition for the topological conjugacy,which helps us construct counter examples. We also give a sufficient condition as well as a method of constructing the topological conjugacy.
基金supported by the National Natural Science Foundation of China (No. 10771132)SGRC (No.GZ 310)the Research Grant of Shanghai University and the Shanghai Leading Academic Discipline Project (No. J50101).
文摘Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.
基金Project supported by the Natural Science Foundation of Tsinghua University
文摘This note will give some classification results for local and global C^1 diffeomorphisms of R under C^1 conjugacy. For γ=1, 2,…, ∞, ω, let D~γ(0)={f: R→R is C~γ; f’(x)】0, x∈R; and f has 0 asits unique fixed point} (local diffeomorphisms of R). In Refs. [1] and [2], we have considered the existence of smooth embedding flows and the other related problems for f∈D~γ(0). As a result, we prove that the following classifications have some numerical invariants
基金the 973 Project Foundation of China (Grant No. TG1999075102)
文摘Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E_n(G)of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E_n(G) are also given.
文摘In this paper,a topological classification of piecewise monotone functions(abbreviated as PM functions)with nonmonotonicity height equal to 1 which are strictly increasing on their characteristic intervals and have finitely many fixed points is presented.
基金supported by National Natural Science Foundation of China(Grant No.11301218)the Nature Science Fund of Shandong Province(Grant No.ZR2014AM020)+4 种基金University of Jinan Research Funds for Doctors(Grant Nos.XBS1335 and XBS1336)the Valencian GovernmentProyecto PROMETEO/2011/30the Spanish GovernmentProyecto(Grant No.MTM2010-19938-C03-02)
文摘Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We show that if 4 divides no A-invariant conjugacy class size of G, then G is solvable. We also characterize the A-invariant structure of G under certain arithmetical conditions on the set of A-invariant class sizes of G by means of the fixed point subgroup, some of which imply the solvability of G. Thus, we extend, for coprime action, several results appeared in the literature on class sizes.
文摘Let G be a group,and let →/ΓC(G) be a digraph whose vertices are the nontrivial conjugacy classes of G and there is an arc from a vertex C to a vertex C' if and only if C≠C' and(C)■(C').In this paper,we characterize finite groups G whose associated digraphs →/ΓC(G) are oriented trees.
文摘An element x of a finite group G is said to be primary if the order of x is a prime power.We define csp2(G)as follows:if|x^(G)|is a prime power for every primary element x of G,where x^(G) is the conjugacy class of x in G,then csp2(G)=0;if there exists a primary element x in G such that|x^(G)|is divisible by at least two distinct primes,then csp2(G)=max{|x^(G)||x∈Gis primary,|x^(G)|is divisible by at least two distinct primes}.In this paper we discuss the influence of the number csp2(G)on the structure of G.
基金partially supported by the National Natural Science Foundation of China(11901169)the Youth Science Foundation of Henan Normal University(2019QK02)the Project for Graduate Education Reform and Quality Improvement of Henan Province and Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control,College of Mathematics and Information Science.
文摘Let G be a finite group and N a normal subgroup of G.Denote by Γ_(G)(N)the graph whose vertices are all distinct G-conjugacy class sizes of non-central elements in N,and two vertices of Γ_(G)(N)are adjacent if and only if they are not coprime numbers.We prove that if the center Z(N)=Z(G)∩N and Γ_(G)(N)is k-regular for k≥1,then either a section of Nis a quasi-Frobenius group or Γ_(G)(N)is a complete graph with k+1 vertices.
基金This work was supported by First-Class Disciplines Foundation of Ningxia Hui Autonomous Region(No.NXYLXK2017B09)the National Natural Science Foundation of China(Nos.11601012,11861002,71771030)+3 种基金the Key Project of North Minzu University(No.ZDZX201804)Natural Science Foundation of Ningxia Hui Autonomous Region(Nos.NZ17103,2018AAC03253)Natural Science Foundation of Guangxi Zhuang Autonomous Region(No.2018GXNSFAA138169)Guangxi Key Laboratory of Cryptography and Information Security(No.GCIS201708).
文摘In this paper,an adaptive three-term conjugate gradient method is proposed for solving unconstrained problems,which generates sufficient descent directions at each iteration.Different from the existent methods,a dynamical adjustment between Hestenes–Stiefel and Dai–Liao conjugacy conditions in our proposed method is developed.Under mild condition,we show that the proposed method converges globally.Numerical experimentation with the new method indicates that it efficiently solves the test problems and therefore is promising.
基金supported by the National Science Foundation(No.0900985)the National Security Agency(No.H98230-13-1-0209)+1 种基金the National Science Foundation(No.DMS-0757722)the Simons Foundation collaboration grant
文摘Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for etale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.
文摘Nowadays some promising authenticated group key agreement protocols are constructed on braid groups, dynamic groups, pairings and bilinear pairings. Hence the non-abelian structure has attracted cryptographers to construct public-key cryptographic protocols. In this article, we propose a new authenticated group key agreement protocol which works in non-abelian near-rings. We have proved that our protocol meets the security attributes under the assumption that the twist conjugacy search problem（TCSP） is hard in near-ring.
基金supported by National Natural Science Foundation of China(Grant Nos.11201401 and 11101258)National Science Foundation for Postdoctoral Scientists of China(Grant No.20100480582)University of Jinan Research Funds for Doctors(Grant Nos.XBS1335 and XBS1336)
文摘Let m, n 〉 1 be two coprime integers. In this paper, we prove that a finite solvable group is nilpotent if the set of the conjugacy class sizes of its primary and biprimary elements is {1, rn, n, mn}.
基金Supported by the Youth Project of Hubei Provincial Department of Education (Q20112807)the Outstanding Young Team Project of Hubei Provincial Higher School (T201009)
文摘Let G be a finite group and π be a set of primes including at least two elements. We write cd（G） and cs（G） to denote the set of complex irreducible character degrees and conjugacy class sizes of G , respectively, and write π（m）to denote the set of all prime divisors of a positive integer m . For any 1≠m∈cd（G） and 1≠m∈cs（G）, in this note, we shall present the corresponding group structures of finite group G in the case π（m）=π , respectively, which generalizes the result of finite groups with character degrees of two distinct primes. Furthermore, we shall see that the influence of the two sets on the group structure is analogous.
