Let G be a finite group and H a subgroup of G.The normal index of H in G is defined as the order of K/H_(G),where K is a normal supplement of H in G such that|K|is minimal and H_(G)≤K■G.Let p be a prime which divide...Let G be a finite group and H a subgroup of G.The normal index of H in G is defined as the order of K/H_(G),where K is a normal supplement of H in G such that|K|is minimal and H_(G)≤K■G.Let p be a prime which divides the order of a group G.In this paper,some characterizations of G being p-solvable or p-supersolvable were obtained by analyzing the normal index of certain subgroups of G.These results can be viewed as local version of recent results in the literature.展开更多
The rapid development of mega low earth orbit(LEO)satellite networks is expected to have a significant impact on 6G networks.Unlike terrestrial networks,due to the high-speed movement of satellites,users will frequent...The rapid development of mega low earth orbit(LEO)satellite networks is expected to have a significant impact on 6G networks.Unlike terrestrial networks,due to the high-speed movement of satellites,users will frequently hand over between satellites even if their positions remain unchanged.Furthermore,the extensive coverage characteristic of satellites leads to massive users executing handovers simultaneously.To address these challenges,we propose a novel double grouping-based group handover scheme(DGGH)specifically tailored for mega LEO satellite networks.First,we develop a user grouping strategy based on beam-limited hierarchical clustering to divide users into distinct groups.Next,we reframe the challenge of managing multiple users’simultaneous handovers as a single-objective optimization problem,solving it with a satellite grouping strategy that leverages the accuracy of greedy algorithms and the simplicity of dynamic programming.Additionally,we develop a group handover algorithm based on minimal handover waiting time to improve the satellite grouping process further.The detailed steps of the DGGH scheme’s handover procedure are meticulously outlined.Comprehensive simulations show that the proposed DGGH scheme outperforms single-user handover schemes in terms of handover signaling over-head and handover success rate.展开更多
Let G be a finite group.A generating set X of G is said to be minimal if no proper subset of X generates G.Let d(G)and m(G)denote the smallest size and the largest size of a minimal generating set of G,respectively.In...Let G be a finite group.A generating set X of G is said to be minimal if no proper subset of X generates G.Let d(G)and m(G)denote the smallest size and the largest size of a minimal generating set of G,respectively.In this paper we present a characterization for finite solvable groups G such that m(G)-d(G)=1 and m(G)≥m(G/N)+m(N)for any non-trivial normal subgroup N of G.展开更多
In this paper,we introduce the set of maximal subgroups with non-trivial core and their corresponding second maximal subgroups.A correlation characterization of the group class J_(pr)is presented by establishing the r...In this paper,we introduce the set of maximal subgroups with non-trivial core and their corresponding second maximal subgroups.A correlation characterization of the group class J_(pr)is presented by establishing the relationship between the core and the second maximal subgroups in these classifications.展开更多
Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG（H）∩ H^g ≤ H for all g C G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = ...Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG（H）∩ H^g ≤ H for all g C G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H N K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature.展开更多
Let G be a group.The family of all sets which are closed in every Hausdorf group topology of G form the family of closed sets of a T_(1) topology M_(G) on G called the Markov topology.Similarly,the family of all algeb...Let G be a group.The family of all sets which are closed in every Hausdorf group topology of G form the family of closed sets of a T_(1) topology M_(G) on G called the Markov topology.Similarly,the family of all algebraic subsets of G forms a family of closed sets for another T_(1)topology Z_(G) on G called the Zarski topology.A subgroup H of G is said to be Markov(resp.Zarski)embedded if the equality M_(G|H)=M_(H)(resp.Z_(G|H)=Z_(H))holds.I's proved that an abirary subgroup of a free group is both Zariski and Markov embedded in it.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
Solar-driven CO_(2)conversion and pollutant removal with an S-scheme heterojunction provides promising approach to alleviate energy shortage and environmental crisis,yet the comprehensive regulation of the charge sepa...Solar-driven CO_(2)conversion and pollutant removal with an S-scheme heterojunction provides promising approach to alleviate energy shortage and environmental crisis,yet the comprehensive regulation of the charge separation and the activation sites of reactant molecules remains challenging.Herein,a dual-active groups regulated S-scheme heterojunction for hydroxy-regulated BiOBr modified amino-functionalized g-C_(3)N_(4)(labeled as HBOB/ACN)was designed by spatially separated dual sites with hydroxyl group(OH)and amino group(NH_(2))toward simultaneously photocatalytic CO_(2)reduction and ciprofloxacin(CIP)oxidation.The optimized HBOB/ACN delivers around 2.74-fold CO yield rate and 1.61-times CIP removal rate in comparison to BiOBr/g-C_(3)N_(4)(BOB/CN)without surface groups,which chiefly ascribed the synergistic effect of OH and NH_(2)group.A series of experiments and theoretical calculation unveiled that the OH and NH_(2)group trapped holes and electrons to participate in CIP oxidation and CO_(2)reduction,respectively.Besides,dual-functional coupled reaction system realized the complete utilization of carriers.This work affords deep insights for dual-group modified S-scheme heterojunctions with redox active sites toward dual-functional coupled reaction system for environment purification and solar fuel production.展开更多
Objectives:In this study,we examined the psychological impact of hierarchical and authoritarian structures in elite sports teams in South Korea on the ego formation and mental health of athletes.We aimed to analyze ho...Objectives:In this study,we examined the psychological impact of hierarchical and authoritarian structures in elite sports teams in South Korea on the ego formation and mental health of athletes.We aimed to analyze how these environments shape psychological well-being in athletes,drawing on Freud’s group psychology theory,while integrating perspectives from the Self-DeterminationTheory(SDT)and Social Identity Theory(SIT).Methods:We applied a qualitative case-study approach,with data collected through in-depth interviews with eight retired elite table tennis players from South Korea.These athletes shared their experiences with psychological mechanisms in their teams and how those mechanisms impacted their mental health throughout their careers.We analyzed the collected data using thematic analysis.Results:The investigated psychological mechanisms significantly influenced the ego development and psychological well-being in athletes.In hierarchical and authoritarian environments,identification and ego idealization suppressed the autonomy of the athletes.Hierarchical order and obedience to authority figures exert significant pressure on athletes to conform,impeding the development of an independent ego.Furthermore,group pressure exacerbates ego erosion and psychological stress,while corporal punishment reinforces psychological pressure and hinders ego integrity.The integration of the SDT highlights the need for autonomy-supportive environments to mitigate such risks,while the SIT emphasizes balancing individual identity with group cohesion to address identity conflicts and psychological tension.Conclusion:Hierarchical structures and authoritarian dynamics in South Korean elite sports teams critically suppress autonomy in athletes,leading to psychological distress and identity conflicts.This study highlights the urgent need for systemic interventions to reform coaching practices and foster autonomysupportive environments.These results contribute to the global discourse on the mental health of athletes in elite sports and offer actionable strategies for improving long-term well-being.Finally,this study provides practical recommendations adaptable to diverse cultural and sporting contexts,enhancing athlete mental health worldwide.展开更多
We find the necessary and sufficient conditions on a coproduct of connected acts over a semigroup to be strongly hopfian. From this, we deduce the conditions of the strong hopfness for unitary acts over groups. Moreov...We find the necessary and sufficient conditions on a coproduct of connected acts over a semigroup to be strongly hopfian. From this, we deduce the conditions of the strong hopfness for unitary acts over groups. Moreover, we prove that a finite coproduct of strongly hopfian acts over an arbitrary semigroup is strongly hopfian.展开更多
With the transformation and upgrading of industries,along with the rapid advancement of artificial intelligence(AI)technologies,higher vocational education in art and design is facing new developmental challenges.As a...With the transformation and upgrading of industries,along with the rapid advancement of artificial intelligence(AI)technologies,higher vocational education in art and design is facing new developmental challenges.As an emerging integrative ecosystem,the municipal industry-education consortium offers a strategic pathway to address these challenges by deepening collaboration between education and industry and fostering the coordinated development of both sectors.Grounded in the framework of the municipal industry-education consortium,this study proposes a top-level organizational model for art and design specialty groups characterized by industry-chain-driven development,interdisciplinary integration,and growth-employment orientation.It further explores multiple development pathways,including innovations in consortium council governance,dynamic talent cultivation models,shared curriculum systems,dual-track teaching teams,and multi-tiered practice bases.Collectively,these pathways establish a comprehensive specialty group system in which core specialties drive development,supporting specialties ensure interdependence,and extended specialties foster mutual advancement.The findings provide theoretical insights and practical guidance for advancing the modernization and high-quality transformation of vocational art and design education,contributing to the cultivation of innovative and application-oriented professionals aligned with industrial restructuring and the development of new productive forces.展开更多
Structural mapping is an important method for studying algebraic structures.Hom-algebra and monoidal Hom-group are new structures produced by algebra and group structural mappings,respectively.These structures are imp...Structural mapping is an important method for studying algebraic structures.Hom-algebra and monoidal Hom-group are new structures produced by algebra and group structural mappings,respectively.These structures are important algebra and group generalizations and are closely related to them.Let(A,β)be a Hom-algebra and(G,α)a monoidal Hom-group.A structure of(A,β)graded by(G,α)is introduced;this structure is called Hom-group graded algebra.This study presents the definition of Hom group graded algebra,provides some examples,and dis-cusses its basic properties.Furthermore,a sufficient and necessary condition that makes(A,β)a strongly(G,α)-graded algebra is explored using a structure mapβand unit 1A.Finally,by using different maps,two sufficient and necessary conditions for a Hom-algebra to be a(G,α)-graded algebra are expressed in different ways.展开更多
Let P be a Sylow p-subgroup of a group G with the smallest generator number d, where p is a prime. Denote by Md（P） = （P1, P2,..., Pd） a set of maximal subgroups of P such that φ（P） = ∩n^d=1Pn. In this paper, w...Let P be a Sylow p-subgroup of a group G with the smallest generator number d, where p is a prime. Denote by Md（P） = （P1, P2,..., Pd） a set of maximal subgroups of P such that φ（P） = ∩n^d=1Pn. In this paper, we investigate the structure of a finite group G under the assumption that the maximal subgroups in Md（P） are weakly s-permutably embedded in G, some interesting results are obtained which generalize some recent results. Finally, we give some further results in terms of weakly s-permutably embedded subgroups.展开更多
Let G be a finite group and N(G)be its norm.Then N(G)is a characteristic subgroup of G which normalizes every subgroup of G.In this paper,we will study the structure of G under one of the following conditions:1)norm q...Let G be a finite group and N(G)be its norm.Then N(G)is a characteristic subgroup of G which normalizes every subgroup of G.In this paper,we will study the structure of G under one of the following conditions:1)norm quotient group G/N(G)is cyclic;2)all Sylow subgroups of G/N(G)are cyclic and in particular if the order of G/N(G)is a square-free number.展开更多
In this paper, we prove that if a torsion nilpotent group G is a weak semi-radicable group, then every Sylow p-group Gp is a central-by-finite p-group, and moreover Gp's center ζ(GP) satisfies |ζ(GP) : (ζ(GP))P...In this paper, we prove that if a torsion nilpotent group G is a weak semi-radicable group, then every Sylow p-group Gp is a central-by-finite p-group, and moreover Gp's center ζ(GP) satisfies |ζ(GP) : (ζ(GP))P| <∞, that is, ζ(GP) = D×F, where D is a divisible Abelian group, and F is a finite Abelian group.展开更多
Let H be a subgroup of a finite group G and p a prime divisor dividing the order of G.We say H is cp-supplemented in G if there exists a supplement T to H in G containing HGsuch that H∩T/HGis a p’-group,where HGis t...Let H be a subgroup of a finite group G and p a prime divisor dividing the order of G.We say H is cp-supplemented in G if there exists a supplement T to H in G containing HGsuch that H∩T/HGis a p’-group,where HGis the core of H in G.A CSp-group is a group in which every p-subgroup is cp-supplemented.In this paper,we characterize the p-solvability and psupersolvability of groups G with some certain p-subgroups being cp-supplemented.Furthermore,we give some equivalent conditions of CSp-group in p-solvable universe.Finally,we give some criteria of CSp-groups for the direct product of two CSp-groups.Our results extend some recent conclusions.展开更多
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
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
基金Supported by the National Natural Science Foundation of China(Grant No.12071092)Guangdong Basic and Applied Basic Research Foundation(Grant No.2025A1515012072)+1 种基金the Natural Science Research Project of Anhui Educational Committee(Grant No.2024AH051298)the Scientific Research Foundation of Bozhou University(Grant No.BYKQ202419).
文摘Let G be a finite group and H a subgroup of G.The normal index of H in G is defined as the order of K/H_(G),where K is a normal supplement of H in G such that|K|is minimal and H_(G)≤K■G.Let p be a prime which divides the order of a group G.In this paper,some characterizations of G being p-solvable or p-supersolvable were obtained by analyzing the normal index of certain subgroups of G.These results can be viewed as local version of recent results in the literature.
基金supported in part by the State Key Laboratory of Micro-Spacecraft Rapid Design and Intelligent Cluster(No.MS01240103)the National Natural Science Foundation of China(No.62071146,No.62431009)+1 种基金the National 2011 Collaborative Innovation Center of Wireless Communication Technologies(No.2242022k60006)the Research Project Fund of Songjiang Laboratory(No.SL20230104).
文摘The rapid development of mega low earth orbit(LEO)satellite networks is expected to have a significant impact on 6G networks.Unlike terrestrial networks,due to the high-speed movement of satellites,users will frequently hand over between satellites even if their positions remain unchanged.Furthermore,the extensive coverage characteristic of satellites leads to massive users executing handovers simultaneously.To address these challenges,we propose a novel double grouping-based group handover scheme(DGGH)specifically tailored for mega LEO satellite networks.First,we develop a user grouping strategy based on beam-limited hierarchical clustering to divide users into distinct groups.Next,we reframe the challenge of managing multiple users’simultaneous handovers as a single-objective optimization problem,solving it with a satellite grouping strategy that leverages the accuracy of greedy algorithms and the simplicity of dynamic programming.Additionally,we develop a group handover algorithm based on minimal handover waiting time to improve the satellite grouping process further.The detailed steps of the DGGH scheme’s handover procedure are meticulously outlined.Comprehensive simulations show that the proposed DGGH scheme outperforms single-user handover schemes in terms of handover signaling over-head and handover success rate.
基金Supported by China Scholarship Council(Grant No.202208360148)the National Natural Science Foundation of China(Grant Nos.12126415,12261042,12301026)the Natural Science Foundation of Jiangxi Province(Grant No.20232BAB211006).
文摘Let G be a finite group.A generating set X of G is said to be minimal if no proper subset of X generates G.Let d(G)and m(G)denote the smallest size and the largest size of a minimal generating set of G,respectively.In this paper we present a characterization for finite solvable groups G such that m(G)-d(G)=1 and m(G)≥m(G/N)+m(N)for any non-trivial normal subgroup N of G.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1237101812201236)+1 种基金the Fundamental Research Funds for the Central Universities(Grant No.B240201093/2013)the Natural Science Foundation of the Anhui Higher Education Institutions(Grant No.2022AH051907)。
文摘In this paper,we introduce the set of maximal subgroups with non-trivial core and their corresponding second maximal subgroups.A correlation characterization of the group class J_(pr)is presented by establishing the relationship between the core and the second maximal subgroups in these classifications.
基金supported by the Deanship of Scientific Research(DSR) at King Abdulaziz University(KAU) represented by the Unit of Research Groups through the grant number(MG/31/01) for the group entitled "Abstract Algebra and its Applications"
文摘Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG（H）∩ H^g ≤ H for all g C G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H N K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature.
基金Supported by the National Natural Science Foundation of China (Grant No. 11071150)the Natural Science Foundation of Shanxi Province (Grant No. 2012011001-3)Shanxi Scholarship Council of China (Grant No. [2011]8-59)
文摘We classify up to isomorphism those finite p-groups, for odd primes p, which contain a cyclic subgroup of index p3.
基金Supported by the Grant-in-Aid for Scientific Research(C)by the Japan Society for the Promotion of Science(20K03615)。
文摘Let G be a group.The family of all sets which are closed in every Hausdorf group topology of G form the family of closed sets of a T_(1) topology M_(G) on G called the Markov topology.Similarly,the family of all algebraic subsets of G forms a family of closed sets for another T_(1)topology Z_(G) on G called the Zarski topology.A subgroup H of G is said to be Markov(resp.Zarski)embedded if the equality M_(G|H)=M_(H)(resp.Z_(G|H)=Z_(H))holds.I's proved that an abirary subgroup of a free group is both Zariski and Markov embedded in it.
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.
文摘Solar-driven CO_(2)conversion and pollutant removal with an S-scheme heterojunction provides promising approach to alleviate energy shortage and environmental crisis,yet the comprehensive regulation of the charge separation and the activation sites of reactant molecules remains challenging.Herein,a dual-active groups regulated S-scheme heterojunction for hydroxy-regulated BiOBr modified amino-functionalized g-C_(3)N_(4)(labeled as HBOB/ACN)was designed by spatially separated dual sites with hydroxyl group(OH)and amino group(NH_(2))toward simultaneously photocatalytic CO_(2)reduction and ciprofloxacin(CIP)oxidation.The optimized HBOB/ACN delivers around 2.74-fold CO yield rate and 1.61-times CIP removal rate in comparison to BiOBr/g-C_(3)N_(4)(BOB/CN)without surface groups,which chiefly ascribed the synergistic effect of OH and NH_(2)group.A series of experiments and theoretical calculation unveiled that the OH and NH_(2)group trapped holes and electrons to participate in CIP oxidation and CO_(2)reduction,respectively.Besides,dual-functional coupled reaction system realized the complete utilization of carriers.This work affords deep insights for dual-group modified S-scheme heterojunctions with redox active sites toward dual-functional coupled reaction system for environment purification and solar fuel production.
文摘Objectives:In this study,we examined the psychological impact of hierarchical and authoritarian structures in elite sports teams in South Korea on the ego formation and mental health of athletes.We aimed to analyze how these environments shape psychological well-being in athletes,drawing on Freud’s group psychology theory,while integrating perspectives from the Self-DeterminationTheory(SDT)and Social Identity Theory(SIT).Methods:We applied a qualitative case-study approach,with data collected through in-depth interviews with eight retired elite table tennis players from South Korea.These athletes shared their experiences with psychological mechanisms in their teams and how those mechanisms impacted their mental health throughout their careers.We analyzed the collected data using thematic analysis.Results:The investigated psychological mechanisms significantly influenced the ego development and psychological well-being in athletes.In hierarchical and authoritarian environments,identification and ego idealization suppressed the autonomy of the athletes.Hierarchical order and obedience to authority figures exert significant pressure on athletes to conform,impeding the development of an independent ego.Furthermore,group pressure exacerbates ego erosion and psychological stress,while corporal punishment reinforces psychological pressure and hinders ego integrity.The integration of the SDT highlights the need for autonomy-supportive environments to mitigate such risks,while the SIT emphasizes balancing individual identity with group cohesion to address identity conflicts and psychological tension.Conclusion:Hierarchical structures and authoritarian dynamics in South Korean elite sports teams critically suppress autonomy in athletes,leading to psychological distress and identity conflicts.This study highlights the urgent need for systemic interventions to reform coaching practices and foster autonomysupportive environments.These results contribute to the global discourse on the mental health of athletes in elite sports and offer actionable strategies for improving long-term well-being.Finally,this study provides practical recommendations adaptable to diverse cultural and sporting contexts,enhancing athlete mental health worldwide.
文摘We find the necessary and sufficient conditions on a coproduct of connected acts over a semigroup to be strongly hopfian. From this, we deduce the conditions of the strong hopfness for unitary acts over groups. Moreover, we prove that a finite coproduct of strongly hopfian acts over an arbitrary semigroup is strongly hopfian.
基金General Project of the 2023 Annual Scheme for Shenzhen’s Education Science“14th Five-Year”Plan:Construction of Art and Design Specialty Groups in Higher Vocational Education Based on the Municipal Industry-Education Consortium(yb23018)2024 University-Level Teaching Achievement Award Cultivation Project of Shenzhen Polytechnic University:“Innovation and Implementation of a Sustainable Talent Development Model Based on Municipal Industry-Education Collaboration and Art-Technology Integration”。
文摘With the transformation and upgrading of industries,along with the rapid advancement of artificial intelligence(AI)technologies,higher vocational education in art and design is facing new developmental challenges.As an emerging integrative ecosystem,the municipal industry-education consortium offers a strategic pathway to address these challenges by deepening collaboration between education and industry and fostering the coordinated development of both sectors.Grounded in the framework of the municipal industry-education consortium,this study proposes a top-level organizational model for art and design specialty groups characterized by industry-chain-driven development,interdisciplinary integration,and growth-employment orientation.It further explores multiple development pathways,including innovations in consortium council governance,dynamic talent cultivation models,shared curriculum systems,dual-track teaching teams,and multi-tiered practice bases.Collectively,these pathways establish a comprehensive specialty group system in which core specialties drive development,supporting specialties ensure interdependence,and extended specialties foster mutual advancement.The findings provide theoretical insights and practical guidance for advancing the modernization and high-quality transformation of vocational art and design education,contributing to the cultivation of innovative and application-oriented professionals aligned with industrial restructuring and the development of new productive forces.
基金The National Natural Science Foundation of China (No. 12271089, 12471033)。
文摘Structural mapping is an important method for studying algebraic structures.Hom-algebra and monoidal Hom-group are new structures produced by algebra and group structural mappings,respectively.These structures are important algebra and group generalizations and are closely related to them.Let(A,β)be a Hom-algebra and(G,α)a monoidal Hom-group.A structure of(A,β)graded by(G,α)is introduced;this structure is called Hom-group graded algebra.This study presents the definition of Hom group graded algebra,provides some examples,and dis-cusses its basic properties.Furthermore,a sufficient and necessary condition that makes(A,β)a strongly(G,α)-graded algebra is explored using a structure mapβand unit 1A.Finally,by using different maps,two sufficient and necessary conditions for a Hom-algebra to be a(G,α)-graded algebra are expressed in different ways.
基金the National Natural Science Foundation of China(Grant Nos.1120108211171353)+2 种基金China Postdoctoral Science Foundation(Grant No.2013T60866)the Natural Science Foundation of Guangdong Province(Grant No.S201204007267)Outstanding Young Teachers Training Project of Guangdong Province(GrantNo.Yq2013061)
文摘Let P be a Sylow p-subgroup of a group G with the smallest generator number d, where p is a prime. Denote by Md（P） = （P1, P2,..., Pd） a set of maximal subgroups of P such that φ（P） = ∩n^d=1Pn. In this paper, we investigate the structure of a finite group G under the assumption that the maximal subgroups in Md（P） are weakly s-permutably embedded in G, some interesting results are obtained which generalize some recent results. Finally, we give some further results in terms of weakly s-permutably embedded subgroups.
基金Supported by the National Natural Science Foundation of China(Grant No.11661023)the Service Industry Development Guide of Guizhou Province(Grant No.Qian Fa Gai Fu Wu[2018]1181)。
文摘Let G be a finite group and N(G)be its norm.Then N(G)is a characteristic subgroup of G which normalizes every subgroup of G.In this paper,we will study the structure of G under one of the following conditions:1)norm quotient group G/N(G)is cyclic;2)all Sylow subgroups of G/N(G)are cyclic and in particular if the order of G/N(G)is a square-free number.
文摘In this paper, we prove that if a torsion nilpotent group G is a weak semi-radicable group, then every Sylow p-group Gp is a central-by-finite p-group, and moreover Gp's center ζ(GP) satisfies |ζ(GP) : (ζ(GP))P| <∞, that is, ζ(GP) = D×F, where D is a divisible Abelian group, and F is a finite Abelian group.
基金Supported by the National Natural Science Foundation of China(Grant No.12071092)the Science and Technology Program of Guangzhou Municipality(Grant No.201804010088).
文摘Let H be a subgroup of a finite group G and p a prime divisor dividing the order of G.We say H is cp-supplemented in G if there exists a supplement T to H in G containing HGsuch that H∩T/HGis a p’-group,where HGis the core of H in G.A CSp-group is a group in which every p-subgroup is cp-supplemented.In this paper,we characterize the p-solvability and psupersolvability of groups G with some certain p-subgroups being cp-supplemented.Furthermore,we give some equivalent conditions of CSp-group in p-solvable universe.Finally,we give some criteria of CSp-groups for the direct product of two CSp-groups.Our results extend some recent conclusions.
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.
