All parabolic subgroups and Borel subgroups of PΩ（2m ＋ 1, F） over a linearable field F of characteristic 0 are shown to be complete groups, provided m 〉 3.
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 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
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
1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal...1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal groups over integral domains have been achieved. Refer to O’Meara, Hahn for example. B. R. McDonald, in [12], determined the automorphisms of O(V) over local rings with 2 a unit by using involutions.展开更多
Let G be a finite group.A subgroup H of G is said to be σ-c-propermutable in G if G has a subgroup B such that G=N_(G)(H)B and for every Hall σ_(i)-subgroup B_(i) of B,there exists an element x∈B such that HB_(i)^(...Let G be a finite group.A subgroup H of G is said to be σ-c-propermutable in G if G has a subgroup B such that G=N_(G)(H)B and for every Hall σ_(i)-subgroup B_(i) of B,there exists an element x∈B such that HB_(i)^(x)=B_(i)^(x) H.In this paper,the influence of σ-c-propermutable subgroups on the structure of finite groups is investigated,and some criteria for a normal subgroup of G to be hypercyclically embedded in G are derived.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
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
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
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
Given a maximal subgroup M of a group G, a θ*-completion C of M is called an s*-completion if either C = G or there exists a subgroup D of G which is not a θ*-completion of M such that D contains C as a maximal subg...Given a maximal subgroup M of a group G, a θ*-completion C of M is called an s*-completion if either C = G or there exists a subgroup D of G which is not a θ*-completion of M such that D contains C as a maximal subgroup. In this paper, we obtain several results on s*-completions which imply G to be solvable or supersolvable.展开更多
In this paper, we generate the wreath product L2 （1 1） wrM12 using only two permutations. Also, we show the structure of some groups containing the wreath product L2（1 1）wrM12. The structure of the groups founded ...In this paper, we generate the wreath product L2 （1 1） wrM12 using only two permutations. Also, we show the structure of some groups containing the wreath product L2（1 1）wrM12. The structure of the groups founded is determined in terms of wreath product （L2 （11）wrM12）wrCt. Some related cases are also included. Also, we will show that S132K＋1 and A132K＋l can be generated using the wreath product （L2 （1 1）wrM12） wr Ck and a transposition in S132K＋1 and an element of order 3 in A132K＋l. We will also show that S132K＋1 and A132K＋1 can be generated using the wreath product L2 （1 1） wrMl2 and an element of order k ＋ 1.展开更多
The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case ...The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.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
In Paolini and Shelah(2024),we proved that the space of countable torsion-free abelian groups is Borel complete.In this paper,we show that our construction from Paolini and Shelah(2024)satisfies several additional pro...In Paolini and Shelah(2024),we proved that the space of countable torsion-free abelian groups is Borel complete.In this paper,we show that our construction from Paolini and Shelah(2024)satisfies several additional properties of interest.We deduce from this that countable torsion-free abelian groups are faithfully Borel complete;in fact,more strongly,we can L_(ω1,ω)-interpret countable graphs in them.Secondly,we show that the relation of pure embeddability(i.e.,elementary embeddability)among countable models of Th(Z^((ω)))is a complete analytic quasi-order.展开更多
Let T_(n) and S_(n) be the full transformation semigroup and the symmetric group on X_(n)={1,2,...,n},respectively.Let G be a transitiveimprimitive subgroupof S_(n) with nontrivial blocksΔand letαbe a transformation...Let T_(n) and S_(n) be the full transformation semigroup and the symmetric group on X_(n)={1,2,...,n},respectively.Let G be a transitiveimprimitive subgroupof S_(n) with nontrivial blocksΔand letαbe a transformation in T_(n)\S_(n).The kernel ofαis the partition of X_(n) induced by the equivalence relation{(x,y)|xα=yα};the kernel type ofαis the partition of n given by the sizes of the parts of the kernel.A transformation semigroup is called synchronizing if it contains a constant map.Then a group G synchronizes a transformationαif the semigroup(G,α)contains a constant map.In this paper,we study a transitive imprimitive permutation group G together with a non-invertible transformationαthat generate a synchronizing semigroup.We mainly discuss 7 cases where G synchronizes a special transformationαwith each kernel class A_(i)(A_(1)j)satisfying|A_(i)∩Δ|=1(|A_(1)j∩Δ|=1)for all blocksΔofG,that is,the kernel type ofαis(|A_(1)|,1,...,1),(|A_(1)1|,...,|A_(1m)|,|A_(2)|,...,|Ar|),or(|A_(1)|,...,|A_(t)|,1,...,1),or the rank is 2,3,4,or n-2.展开更多
As a potential adsorption material,it is still a challenge for activated carbon fiber(ACF)in efficient adsorption of ethanol due to its nonpolar surface,which is mainly emitted from the grain drying industry.This stud...As a potential adsorption material,it is still a challenge for activated carbon fiber(ACF)in efficient adsorption of ethanol due to its nonpolar surface,which is mainly emitted from the grain drying industry.This study prepared surface polarity-modified ACF using the heteroatom doping method.The modified ACF possessed a richer array of strongly polar oxygen/nitrogen-containing functional groups(primarily phenolic hydroxyl and lactone groups),a larger specific surface are1,and a more developed micropore structure.The adsorption capacities of ethanol for O-ACF and N-ACF were 4.110 mmol/g and 1.698 mmol/g,respectively,which were 11.3 times and 4.7 times those of unmodified ACF.This was a significant improvement over our previous work(0.363 mmol/g).The improvement of adsorption capacity for the N-ACF was mainly due to the higher specific surface are1,greater number of micropores(more adsorption sites)and abundant existence of defects,whereas,for O-ACF,the improvement mainly relied on the abundant presence of oxygen-containing functional groups on the surface.However,water had a negative effect on the adsorption of ethanol for the modified ACF due to competitive adsorption and the disappearance of capillary condensation.It was further revealed that the adsorption process of ethanol and water was quite different.It obeyed the linear driving force(LDF)model for ethanol adsorption,however,the intraparticle diffusion(IPD)model for water adsorption.展开更多
文摘All parabolic subgroups and Borel subgroups of PΩ（2m ＋ 1, F） over a linearable field F of characteristic 0 are shown to be complete groups, provided m 〉 3.
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 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.
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.
文摘1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal groups over integral domains have been achieved. Refer to O’Meara, Hahn for example. B. R. McDonald, in [12], determined the automorphisms of O(V) over local rings with 2 a unit by using involutions.
文摘Let G be a finite group.A subgroup H of G is said to be σ-c-propermutable in G if G has a subgroup B such that G=N_(G)(H)B and for every Hall σ_(i)-subgroup B_(i) of B,there exists an element x∈B such that HB_(i)^(x)=B_(i)^(x) H.In this paper,the influence of σ-c-propermutable subgroups on the structure of finite groups is investigated,and some criteria for a normal subgroup of G to be hypercyclically embedded in G are derived.
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,
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.
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.
文摘Given a maximal subgroup M of a group G, a θ*-completion C of M is called an s*-completion if either C = G or there exists a subgroup D of G which is not a θ*-completion of M such that D contains C as a maximal subgroup. In this paper, we obtain several results on s*-completions which imply G to be solvable or supersolvable.
文摘In this paper, we generate the wreath product L2 （1 1） wrM12 using only two permutations. Also, we show the structure of some groups containing the wreath product L2（1 1）wrM12. The structure of the groups founded is determined in terms of wreath product （L2 （11）wrM12）wrCt. Some related cases are also included. Also, we will show that S132K＋1 and A132K＋l can be generated using the wreath product （L2 （1 1）wrM12） wr Ck and a transposition in S132K＋1 and an element of order 3 in A132K＋l. We will also show that S132K＋1 and A132K＋1 can be generated using the wreath product L2 （1 1） wrMl2 and an element of order k ＋ 1.
文摘The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.
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.
基金supported by Project Progetti di Rilevante Interesse Nazionale 2022“Models,Sets and Classifications”,Project 2022TECZJA and Istituto Nazionale di Alta Matematica Project 2024(Consolidator Grant)“Groups,Crystals and Classifications”supported by Israel Science Foundation(Grant Nos.1838/19 and 2320/23).
文摘In Paolini and Shelah(2024),we proved that the space of countable torsion-free abelian groups is Borel complete.In this paper,we show that our construction from Paolini and Shelah(2024)satisfies several additional properties of interest.We deduce from this that countable torsion-free abelian groups are faithfully Borel complete;in fact,more strongly,we can L_(ω1,ω)-interpret countable graphs in them.Secondly,we show that the relation of pure embeddability(i.e.,elementary embeddability)among countable models of Th(Z^((ω)))is a complete analytic quasi-order.
基金Supported by NSFC (No.12401024)the Scientific Research Innovation Project of Lingnan Normal University (Nos.LT2401,LT2410)。
文摘Let T_(n) and S_(n) be the full transformation semigroup and the symmetric group on X_(n)={1,2,...,n},respectively.Let G be a transitiveimprimitive subgroupof S_(n) with nontrivial blocksΔand letαbe a transformation in T_(n)\S_(n).The kernel ofαis the partition of X_(n) induced by the equivalence relation{(x,y)|xα=yα};the kernel type ofαis the partition of n given by the sizes of the parts of the kernel.A transformation semigroup is called synchronizing if it contains a constant map.Then a group G synchronizes a transformationαif the semigroup(G,α)contains a constant map.In this paper,we study a transitive imprimitive permutation group G together with a non-invertible transformationαthat generate a synchronizing semigroup.We mainly discuss 7 cases where G synchronizes a special transformationαwith each kernel class A_(i)(A_(1)j)satisfying|A_(i)∩Δ|=1(|A_(1)j∩Δ|=1)for all blocksΔofG,that is,the kernel type ofαis(|A_(1)|,1,...,1),(|A_(1)1|,...,|A_(1m)|,|A_(2)|,...,|Ar|),or(|A_(1)|,...,|A_(t)|,1,...,1),or the rank is 2,3,4,or n-2.
基金supported by the National Key R&D Program of China(Nos.2022YFB4101500 and 2022YFE0209500)the National Natural Science Foundation of China(Nos.22276191 and 21976177)the Qinghai Province Air Pollution Assessment and Fine Management Support Project,and the University of Chinese Academy of Science.
文摘As a potential adsorption material,it is still a challenge for activated carbon fiber(ACF)in efficient adsorption of ethanol due to its nonpolar surface,which is mainly emitted from the grain drying industry.This study prepared surface polarity-modified ACF using the heteroatom doping method.The modified ACF possessed a richer array of strongly polar oxygen/nitrogen-containing functional groups(primarily phenolic hydroxyl and lactone groups),a larger specific surface are1,and a more developed micropore structure.The adsorption capacities of ethanol for O-ACF and N-ACF were 4.110 mmol/g and 1.698 mmol/g,respectively,which were 11.3 times and 4.7 times those of unmodified ACF.This was a significant improvement over our previous work(0.363 mmol/g).The improvement of adsorption capacity for the N-ACF was mainly due to the higher specific surface are1,greater number of micropores(more adsorption sites)and abundant existence of defects,whereas,for O-ACF,the improvement mainly relied on the abundant presence of oxygen-containing functional groups on the surface.However,water had a negative effect on the adsorption of ethanol for the modified ACF due to competitive adsorption and the disappearance of capillary condensation.It was further revealed that the adsorption process of ethanol and water was quite different.It obeyed the linear driving force(LDF)model for ethanol adsorption,however,the intraparticle diffusion(IPD)model for water adsorption.
