Suppose R is a principal ideal ring, R^* is a multiplicative group which is composed of all reversible elements in R, and Mn(R), GL(n,R), SL(n,R) are denoted by,Mn(R)={A=(aij)n×n|aij∈R,i,j=1,2…,n},G...Suppose R is a principal ideal ring, R^* is a multiplicative group which is composed of all reversible elements in R, and Mn(R), GL(n,R), SL(n,R) are denoted by,Mn(R)={A=(aij)n×n|aij∈R,i,j=1,2…,n},GL(n,R)={g|g∈Mn(R),det g∈R^*},SL(n,R)={g∈GL(n,R)|det g=1},SL(n,R)≤G≤GL(n,R)(n≥3),respectively,then basing on these facts, this paper mainly focus on discussing all extended groups of Gr={(O D^A B)∈G|A∈GL(r,R),(1≤r〈n)}in G when R is a principal ideal ring.展开更多
A typical example for the algebraic groups is the general linear groups G=GL(n,F), we have studied the structure of such groups and paid special attention to its important substructures, namely the Parabolic subgroups...A typical example for the algebraic groups is the general linear groups G=GL(n,F), we have studied the structure of such groups and paid special attention to its important substructures, namely the Parabolic subgroups. For a given G we computed all the Parabolic subgroups and determined their number, depending on the fact that any finite group has a composition series and the composition factors of a composition series are simple groups which are completely classified, we report here some investigations on the computed Parabolic subgroups. This has been done with the utility of GAP.展开更多
Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In...Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.展开更多
Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k fi...Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp(3), where p 〉 3 is a prime number, is at most 2-recognizable.展开更多
Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where...Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn(L) = PLn (L) for any n (n 〉 2), in which GLn(L) is the group of all n × n invertible matrices over L and PLn(L) is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.展开更多
Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time ...Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.展开更多
Unlike the traditional independent component analysis(ICA)algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices,in this paper we ...Unlike the traditional independent component analysis(ICA)algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices,in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening.The new algorithms are based on the special linear group SL(n).In order to achieve our target,we first provide a representation theory for any matrix in SL(n),which only simply uses the product of multiple exponentials of traceless matrices.Based on the matrix representation theory,two novel ICA algorithms are developed along with simple analysis on their equilibrium points.Moreover,we apply our methods to the classical problem of signal separation.The experimental results indicate that the superior convergence of our proposed algorithms,which can be expected as two viable alternatives to the ICA algorithms available in publications.展开更多
For a commutative ring with identity, we obtain a complete description of all overgroups of unitary groups U2nR (n ≥ 5), which include symplectic, ordinary orthogonal and standard unitary groups, in linear group GL2nR.
The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dim...The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G.展开更多
Let R and A be commutative rings with identity, and m and n be integers ≥3. Consider when and how an isomorphism E_m(R)E_n(A) can be lifted to an isomorphism between the corresponding Steinberg groups. It was proved ...Let R and A be commutative rings with identity, and m and n be integers ≥3. Consider when and how an isomorphism E_m(R)E_n(A) can be lifted to an isomorphism between the corresponding Steinberg groups. It was proved that, if E_m(R) is isomorphic to E_n(A) then m=n (cf. Ref. [1]). When n≥4, every isomorphism E_n(R)E_n(A) is of the standard type, and it can be naturally and uniquely lifted to an isomorphism from St_n(R) to St_n(A) (cf. Refs. [1] and [2]). However, the case n=3 is different from that n≥4,展开更多
We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several m...We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-展开更多
Suppose F is a field consisting of at least four elements. Let Mn(F) and SP2n(F) be the linear space of all n × n matrices and the group of all 2n × 2n symplectic matrices over F, respectively. A linear ...Suppose F is a field consisting of at least four elements. Let Mn(F) and SP2n(F) be the linear space of all n × n matrices and the group of all 2n × 2n symplectic matrices over F, respectively. A linear operator L : M2n(F) → M2n(F) is said to preserve the symplectic group if L(SP2n(F)) = SP2n(F). It is shown that L is an invertible preserver of the symplectic group if and only if L takes the form (i) L(X) = QPXP^-1 for any X ∈ M2n(F) or (ii) L(X) = QPX^TP^-1 for any X ∈M2n(F), where Q ∈ SP2n(F) and P is a generalized symplectic matrix. This generalizes the result derived by Pierce in Canad J. Math., 3(1975), 715-724.展开更多
Firstly, in the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and ge...Firstly, in the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced. Secondly, the conclusion that any two nonzero generalized orthogonal groups must be linearly independent group is proven. And the existence of nonzero generalized orthogonal group and its linear correlation are discussed preliminarily, as well as some related properties of nonempty generalized orthogonal group in specific normed linear space namely the <em>l<sub>p</sub></em> space.展开更多
Structures of two classes of solvable subgroups in SL(3, C) are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is di...Structures of two classes of solvable subgroups in SL(3, C) are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is discussed.展开更多
For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G i...For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.展开更多
基金Foundation item: Supported by the National Natural Science Foundation of China(10771093)
文摘Suppose R is a principal ideal ring, R^* is a multiplicative group which is composed of all reversible elements in R, and Mn(R), GL(n,R), SL(n,R) are denoted by,Mn(R)={A=(aij)n×n|aij∈R,i,j=1,2…,n},GL(n,R)={g|g∈Mn(R),det g∈R^*},SL(n,R)={g∈GL(n,R)|det g=1},SL(n,R)≤G≤GL(n,R)(n≥3),respectively,then basing on these facts, this paper mainly focus on discussing all extended groups of Gr={(O D^A B)∈G|A∈GL(r,R),(1≤r〈n)}in G when R is a principal ideal ring.
文摘A typical example for the algebraic groups is the general linear groups G=GL(n,F), we have studied the structure of such groups and paid special attention to its important substructures, namely the Parabolic subgroups. For a given G we computed all the Parabolic subgroups and determined their number, depending on the fact that any finite group has a composition series and the composition factors of a composition series are simple groups which are completely classified, we report here some investigations on the computed Parabolic subgroups. This has been done with the utility of GAP.
基金supported by NSFC(11671063)a grant from the Simons Foundation(#499532 to Yong Yang)a grant from the Simons Foundation(#280770 to Thomas M.Keller).
文摘Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.
基金Supported by the research council of College of Science, the University of Tehran (Grant No. 6103014-1-03)
文摘Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp(3), where p 〉 3 is a prime number, is at most 2-recognizable.
基金National Natural Science Foundation of China (60174013) Research Foundation for Doctoral Program of Higher Education (20020027013)+1 种基金 Science and Technology Key Project Foundation of Ministry of Education (03184) Major State Basic Research Development Program of China (2002CB312200)
文摘Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn(L) = PLn (L) for any n (n 〉 2), in which GLn(L) is the group of all n × n invertible matrices over L and PLn(L) is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.
基金supported by National Natural Science Foundation of China(Grant No.11671063)a grant from the Simons Foundation(Grant No.280770 to Thomas M.Keller)a grant from the Simons Foundation(Grant No.499532 to Yong Yang)。
文摘Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.
文摘Unlike the traditional independent component analysis(ICA)algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices,in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening.The new algorithms are based on the special linear group SL(n).In order to achieve our target,we first provide a representation theory for any matrix in SL(n),which only simply uses the product of multiple exponentials of traceless matrices.Based on the matrix representation theory,two novel ICA algorithms are developed along with simple analysis on their equilibrium points.Moreover,we apply our methods to the classical problem of signal separation.The experimental results indicate that the superior convergence of our proposed algorithms,which can be expected as two viable alternatives to the ICA algorithms available in publications.
基金supported by the National Natural Science Foundation(Grant No.10571033)the Research Fund for the Doctoral of Higher Education of China(Grant No.20040213006)Cultivation Fund of the Key Scientific and Technical Innovation Project Ministry of Education of China(Grant No.704004).
文摘For a commutative ring with identity, we obtain a complete description of all overgroups of unitary groups U2nR (n ≥ 5), which include symplectic, ordinary orthogonal and standard unitary groups, in linear group GL2nR.
基金Tianyuan Mathematics Foundation of NSFC (Grant No.10626050)the Knowledge Innovation Program of the Chinese Academy of Sciences
文摘The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G.
基金Project supported by the National Natural Science Foundation of China
文摘Let R and A be commutative rings with identity, and m and n be integers ≥3. Consider when and how an isomorphism E_m(R)E_n(A) can be lifted to an isomorphism between the corresponding Steinberg groups. It was proved that, if E_m(R) is isomorphic to E_n(A) then m=n (cf. Ref. [1]). When n≥4, every isomorphism E_n(R)E_n(A) is of the standard type, and it can be naturally and uniquely lifted to an isomorphism from St_n(R) to St_n(A) (cf. Refs. [1] and [2]). However, the case n=3 is different from that n≥4,
文摘We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-
文摘Suppose F is a field consisting of at least four elements. Let Mn(F) and SP2n(F) be the linear space of all n × n matrices and the group of all 2n × 2n symplectic matrices over F, respectively. A linear operator L : M2n(F) → M2n(F) is said to preserve the symplectic group if L(SP2n(F)) = SP2n(F). It is shown that L is an invertible preserver of the symplectic group if and only if L takes the form (i) L(X) = QPXP^-1 for any X ∈ M2n(F) or (ii) L(X) = QPX^TP^-1 for any X ∈M2n(F), where Q ∈ SP2n(F) and P is a generalized symplectic matrix. This generalizes the result derived by Pierce in Canad J. Math., 3(1975), 715-724.
文摘Firstly, in the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced. Secondly, the conclusion that any two nonzero generalized orthogonal groups must be linearly independent group is proven. And the existence of nonzero generalized orthogonal group and its linear correlation are discussed preliminarily, as well as some related properties of nonempty generalized orthogonal group in specific normed linear space namely the <em>l<sub>p</sub></em> space.
文摘Structures of two classes of solvable subgroups in SL(3, C) are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is discussed.
基金This work has been supported by the Research Institute for Fundamental Sciences Tabriz,Iran.
文摘For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.