A subgroup H of a group G is called F-z-supplemented in G if there exists a subgroup K of G, such that G = HK and H∩K≤ Z∞F(G), where Z∞F(G) is the F-hypercenter of G. We obtain some results about the F-z-suppl...A subgroup H of a group G is called F-z-supplemented in G if there exists a subgroup K of G, such that G = HK and H∩K≤ Z∞F(G), where Z∞F(G) is the F-hypercenter of G. We obtain some results about the F-z-supplemented subgroups and use them to determine the structure of some groups.展开更多
Let G be a hyper finite locally solvable group, A a minimax ZG-medule, a locally defined formation consisting of locally solvable groups, A has no nonzero infinite irreducible ZG-factors, and G ∈ . The following resu...Let G be a hyper finite locally solvable group, A a minimax ZG-medule, a locally defined formation consisting of locally solvable groups, A has no nonzero infinite irreducible ZG-factors, and G ∈ . The following results are proved: if A has a maximal submodule B such that A/B is , central in G and B has no nonzero central ZG-factors, then A has an decomposition; ifA has an irreducible central submodule B such that all ZG-composition factors of A/B are o^eccentric, then A has an decomposition.展开更多
On the basis of the quasi-isomorphism of finite groups, a new mapping, weak isomorphism, from a finite group to another finite group is defined. Let G and H be two finite groups and G be weak-isomorphic to H. Then G≌...On the basis of the quasi-isomorphism of finite groups, a new mapping, weak isomorphism, from a finite group to another finite group is defined. Let G and H be two finite groups and G be weak-isomorphic to H. Then G≌H if G satisfies one of the following conditions. 1) G is a finite Abelian group. 2) The order of G is p^3. 3 ) The order of G is p^n+1 and G has a cyclic normal subgroup N = 〈a〉 of order p^n. 4) G is a nilpotent group and if p^││G│, then for any P ∈ Sylp (G), P has a cyclic maximal subgroup, where p is a prime; 5) G is a maximal class group of order p4(p〉3).展开更多
For a characterχof a finite group G,the number cod(χ)≔∣G:ker(χ)∣/χ(1)is called the codegree ofχ.In this paper,we give a solvability criterion for a finite group G depending on the minimum of the ratioχ(1)^(2)/...For a characterχof a finite group G,the number cod(χ)≔∣G:ker(χ)∣/χ(1)is called the codegree ofχ.In this paper,we give a solvability criterion for a finite group G depending on the minimum of the ratioχ(1)^(2)/cod(χ),whenχvaries among the irreducible characters of G.展开更多
Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by B_(1)and B_(2),respectively.In this p...Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by B_(1)and B_(2),respectively.In this paper,a characterisation of groups in B_(1)or in B_(2)is given.By applying this new characterisation,some new properties of B_(1)(B_(2))and new proofs of many known results about B_(1)or B_(2)are obtained.Further,closure properties of B_(1)and B_(2)are discussed.展开更多
The spectrum of a finite group is the set of element orders of this group.The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum,in particular,to list all finite si...The spectrum of a finite group is the set of element orders of this group.The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum,in particular,to list all finite simple groups for which the recognition problem is solved.展开更多
For a finite group G,the power graph P(G)is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x and y are adjacent if and only if x^(i)=y or y^(i)=x,for 2≤i,j≤n.In this ...For a finite group G,the power graph P(G)is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x and y are adjacent if and only if x^(i)=y or y^(i)=x,for 2≤i,j≤n.In this paper,we obtain the distance Laplacian spectrum of power graphs of finite groups such as cyclic groups,dihedral groups,dicyclic groups,abelian groups and elementary abelian p groups.Moreover,we find the largest and second smallest distance Laplacian eigenvalue of power graphs of such groups.展开更多
For an irreducible character x of a finite group G,we define its codegree as cod(x)=|G:ker x|/x(1) .In this paper,we introduce some known results x(1)and unsolved problems about character codegrees in finite groups.
Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some nece...Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.展开更多
In this paper we study the influence of the partial cover and avoidance property on the subgroups of some relevant families of subgroups in a finite group.
A subgroup E of a finite group G is called hypercyclically embedded in G if every chief factor of G below E is cyclic.Let A be a subgroup of a group G.Then we call any chief factor H/AG of G a G-boundary factor of A.F...A subgroup E of a finite group G is called hypercyclically embedded in G if every chief factor of G below E is cyclic.Let A be a subgroup of a group G.Then we call any chief factor H/AG of G a G-boundary factor of A.For any G-boundary factor H/AG of A,we call the subgroup(A∩H)/AG of G/AG a G-trace of A.On the basis of these notions,we give some new characterizations of hypercyclically embedded subgroups.展开更多
Thompson's theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson's theorem, a finite group is solvable if it has an abelian maximal...Thompson's theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson's theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.展开更多
Let a = {σi| i ∈ I} be some partition of the set of all primes P, G a finite group and σ(G) = {σi|σi ∩ π (G) ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠...Let a = {σi| i ∈ I} be some partition of the set of all primes P, G a finite group and σ(G) = {σi|σi ∩ π (G) ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-semipermutablc in G with respect to H if HHi x = Hi x H for all x ∈ G and all x ∈ G and all Hi ∈H such that (|H|, |Hi|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.展开更多
Letσ={σi|i∈I}be some partition of the set P of all primes and G afinite group.A set H of subgroups of G is said to be a complete Hallσ-set of G ifevery member≠1 of H is a Hallσi-subgroup of G for some i c l and ...Letσ={σi|i∈I}be some partition of the set P of all primes and G afinite group.A set H of subgroups of G is said to be a complete Hallσ-set of G ifevery member≠1 of H is a Hallσi-subgroup of G for some i c l and H containsexactly one Hallσi-subgroup of G for every i such thatσi∩π(G)≠Ø.A subgroupA of G is said to be H-permutable if A permutes with all members of the completeHallσ-set H of G.In this paper,we study the structure of G under the assuming thatsome subgroups of G areσ-permutable.展开更多
In this paper, we prove the following theorem: Let p be a prime number, P a Sylow psubgroup of a group G and π = π(G) / {p}. If P is seminormal in G, then the following statements hold: 1) G is a p-soluble gro...In this paper, we prove the following theorem: Let p be a prime number, P a Sylow psubgroup of a group G and π = π(G) / {p}. If P is seminormal in G, then the following statements hold: 1) G is a p-soluble group and P' ≤ Op(G); 2) lp(G) ≤ 2 and lπ(G) ≤ 2; 3) if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.展开更多
A subgroup H of a finite group G is called a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H n K is a CAP-subgroup of G. In this paper, we investigate the influence of fewer c#-...A subgroup H of a finite group G is called a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H n K is a CAP-subgroup of G. In this paper, we investigate the influence of fewer c#- normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.展开更多
Let G be a finite group and H a subgroup of G. Recall that H is said to be aTI-subgroup ofG ifHg∩H = 1 or H for each b∈ G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group G...Let G be a finite group and H a subgroup of G. Recall that H is said to be aTI-subgroup ofG ifHg∩H = 1 or H for each b∈ G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group G are TI-subgroups, then G is soluble, and all non-nilpotent subgroups of G are normal.展开更多
Motivated by Problem 164 proposed by Y. Berkovich and E. Zhmud' in their book "Characters of Finite Groups", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This ...Motivated by Problem 164 proposed by Y. Berkovich and E. Zhmud' in their book "Characters of Finite Groups", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character codegree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper.展开更多
文摘A subgroup H of a group G is called F-z-supplemented in G if there exists a subgroup K of G, such that G = HK and H∩K≤ Z∞F(G), where Z∞F(G) is the F-hypercenter of G. We obtain some results about the F-z-supplemented subgroups and use them to determine the structure of some groups.
文摘Let G be a hyper finite locally solvable group, A a minimax ZG-medule, a locally defined formation consisting of locally solvable groups, A has no nonzero infinite irreducible ZG-factors, and G ∈ . The following results are proved: if A has a maximal submodule B such that A/B is , central in G and B has no nonzero central ZG-factors, then A has an decomposition; ifA has an irreducible central submodule B such that all ZG-composition factors of A/B are o^eccentric, then A has an decomposition.
文摘On the basis of the quasi-isomorphism of finite groups, a new mapping, weak isomorphism, from a finite group to another finite group is defined. Let G and H be two finite groups and G be weak-isomorphic to H. Then G≌H if G satisfies one of the following conditions. 1) G is a finite Abelian group. 2) The order of G is p^3. 3 ) The order of G is p^n+1 and G has a cyclic normal subgroup N = 〈a〉 of order p^n. 4) G is a nilpotent group and if p^││G│, then for any P ∈ Sylp (G), P has a cyclic maximal subgroup, where p is a prime; 5) G is a maximal class group of order p4(p〉3).
基金Supported by the National Natural Science Foundation of China(Grant Nos.11971391,12071376,12301018,12171058,12326356)the Natural Science Foundation of Jiangsu Province(Grant No.BK20231356)+1 种基金the Natural Science Foundation for the Universities in Jiangsu Province(Grant No.23KJB110002)The first and second authors are supported by the Chinese Scholarship Council。
文摘For a characterχof a finite group G,the number cod(χ)≔∣G:ker(χ)∣/χ(1)is called the codegree ofχ.In this paper,we give a solvability criterion for a finite group G depending on the minimum of the ratioχ(1)^(2)/cod(χ),whenχvaries among the irreducible characters of G.
基金supported by the project of NSF of China(Grant No.12071092)the major project of Basic and Applied Research(Natural Science)in Guangdong Province,China(Grant No.2017KZDXM058)the Science and Technology Program of Guangzhou Municipality,China(Grant No.201804010088)。
文摘Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by B_(1)and B_(2),respectively.In this paper,a characterisation of groups in B_(1)or in B_(2)is given.By applying this new characterisation,some new properties of B_(1)(B_(2))and new proofs of many known results about B_(1)or B_(2)are obtained.Further,closure properties of B_(1)and B_(2)are discussed.
基金supported by Foreign Experts program in Jiangsu Province(No.JSB2018014)supported by the National Natural Science Foundation of China(No.12171126)+1 种基金supported by the RFBR(No.20-51-00007)supported by the National Natural Science Foundation of China(11171364,11671063).
文摘The spectrum of a finite group is the set of element orders of this group.The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum,in particular,to list all finite simple groups for which the recognition problem is solved.
基金Supported by SERB-DST,New Delhi,under the research project number MTR/2017/000084the third author is supported by NSFC (Grant Nos.11931006 and 11971011)。
文摘For a finite group G,the power graph P(G)is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x and y are adjacent if and only if x^(i)=y or y^(i)=x,for 2≤i,j≤n.In this paper,we obtain the distance Laplacian spectrum of power graphs of finite groups such as cyclic groups,dihedral groups,dicyclic groups,abelian groups and elementary abelian p groups.Moreover,we find the largest and second smallest distance Laplacian eigenvalue of power graphs of such groups.
基金support provided by the National Natural Science Foundation of China(Grant No.12171058).
文摘For an irreducible character x of a finite group G,we define its codegree as cod(x)=|G:ker x|/x(1) .In this paper,we introduce some known results x(1)and unsolved problems about character codegrees in finite groups.
基金National Natural Science Foundation of China(No.11671258)。
文摘Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.
基金Proyecto MTM 2007-68010-C03-02MTM 2007-68010-C03-01,respectively,of Ministerio de Educación y Ciencia of Spain
文摘In this paper we study the influence of the partial cover and avoidance property on the subgroups of some relevant families of subgroups in a finite group.
基金Research of the first author is supported by aNNSFgrant ofChina(Grant#11371335)WuWen-Tsun Key Laboratory of Mathematics,USTC,Chinese Academy of Sciences.Research of the second author supported by Chinese Academy of Sciences Visiting Professorship for Senior International Scientists(Grant No.2010T2J12).
文摘A subgroup E of a finite group G is called hypercyclically embedded in G if every chief factor of G below E is cyclic.Let A be a subgroup of a group G.Then we call any chief factor H/AG of G a G-boundary factor of A.For any G-boundary factor H/AG of A,we call the subgroup(A∩H)/AG of G/AG a G-trace of A.On the basis of these notions,we give some new characterizations of hypercyclically embedded subgroups.
基金Supported by the National Natural Science Foundation of Chinathe Natural Science Foundation of Guangxi Autonomous Region (No.0249001)
文摘For any saturated formation F of finite groups containing all supersolvable groups, the groups in F are characterized by F-abnormal maximal subgroups.
基金Supported by National Natural Science Foundation of China (Grant No. 10871032), China Postdoctoral Science Foundation (Grant No. 20100470136) the second author is supported in part by "Agencija za raziskovalno dejavnost Republike Slovenije", proj. mladi raziskovalci, "Agencija za raziskovalno dejavnost Republike Slovenije", Research Program P1-0285
文摘Thompson's theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson's theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.
基金Supported by NNSF(Grant No.11771409)Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences
文摘Let a = {σi| i ∈ I} be some partition of the set of all primes P, G a finite group and σ(G) = {σi|σi ∩ π (G) ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-semipermutablc in G with respect to H if HHi x = Hi x H for all x ∈ G and all x ∈ G and all Hi ∈H such that (|H|, |Hi|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.
基金a NNSF grant of China(Grant#11371335)Wu Wen-TsunKey Laboratory of Mathematics of Chinese Academy of Sciences.Research of the third author is supported by Chinese Academy of Sciences Visiting Professorship for Senior International Scientists(Grant No.2010T2J12).
文摘Letσ={σi|i∈I}be some partition of the set P of all primes and G afinite group.A set H of subgroups of G is said to be a complete Hallσ-set of G ifevery member≠1 of H is a Hallσi-subgroup of G for some i c l and H containsexactly one Hallσi-subgroup of G for every i such thatσi∩π(G)≠Ø.A subgroupA of G is said to be H-permutable if A permutes with all members of the completeHallσ-set H of G.In this paper,we study the structure of G under the assuming thatsome subgroups of G areσ-permutable.
文摘In this paper, we prove the following theorem: Let p be a prime number, P a Sylow psubgroup of a group G and π = π(G) / {p}. If P is seminormal in G, then the following statements hold: 1) G is a p-soluble group and P' ≤ Op(G); 2) lp(G) ≤ 2 and lπ(G) ≤ 2; 3) if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.
文摘A subgroup H of a finite group G is called a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H n K is a CAP-subgroup of G. In this paper, we investigate the influence of fewer c#- normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.
基金The first author was supported by NSFC (Grant 11201401) and the China Postdoctoral Science Foundation (Grant 201104027). The second author was supported by H.C. Orsted Postdoctoral Fellowship at DTU (Technical University of Denmark).
文摘Let G be a finite group and H a subgroup of G. Recall that H is said to be aTI-subgroup ofG ifHg∩H = 1 or H for each b∈ G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group G are TI-subgroups, then G is soluble, and all non-nilpotent subgroups of G are normal.
文摘Motivated by Problem 164 proposed by Y. Berkovich and E. Zhmud' in their book "Characters of Finite Groups", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character codegree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper.