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
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
In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=...In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.展开更多
Using a new symmetry group theory, the transformation groups and symmetries of the general Broer-Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.
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
Canonical differential calculi are defined for finitely generated Abelian groups with involutions existing consistently. Two such the canonical calculi are presented. Fermionic representations for canonical calculi ar...Canonical differential calculi are defined for finitely generated Abelian groups with involutions existing consistently. Two such the canonical calculi are presented. Fermionic representations for canonical calculi are defined based on quantized calculi. Fermionic representations for aforementioned two canonical calculi are searched out.展开更多
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
Paper considers the calculation of the values of Gibbs derivatives on finite Abelian groups. The calculation procedure is based upon the decision diagram representation of functions defined on finite Abelian groups. A...Paper considers the calculation of the values of Gibbs derivatives on finite Abelian groups. The calculation procedure is based upon the decision diagram representation of functions defined on finite Abelian groups. Approach permits processing of large functions.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
Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The followi...Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained： if G∈F, f（ ∞ ） include f（p）, f（p） ≠φ for each p∈π, and A has no nonzero F central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F central ZG-factors.展开更多
Let Fbe a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A a noetherian ZG-module with all irreducible ZG-factors being finite, G∈F, f(∞)f(p), ...Let Fbe a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A a noetherian ZG-module with all irreducible ZG-factors being finite, G∈F, f(∞)f(p), f(p)≠ for each p∈π. The following conclutions are obtained: (1) if there exists a maximal submodule B of A such that A/B is F-central in G and B has no nonzero F-central ZG-factors, then A has an F-decomposition; (2) if there exists an irreducible F-central submodule B of A such that all ZG-composition factors of A/B are F-ecentric, then A has an F-decomposition.展开更多
Let (?) be a formation locally defined by f(P), G ∈ (?) and A a ZG-module, where p ∈ π = { all primes and symbol ∞}. Then a p-main-factor U/V of G is said to be (?)-central in G if G/CG(U/V) ∈f(p). In this paper,...Let (?) be a formation locally defined by f(P), G ∈ (?) and A a ZG-module, where p ∈ π = { all primes and symbol ∞}. Then a p-main-factor U/V of G is said to be (?)-central in G if G/CG(U/V) ∈f(p). In this paper, we have proved that: let (?) be a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A an artinian ZG-module with all irreducible ZG-factors of A being finite; if G ∈ (?), f(∞) ≡ f(p) . f(p)≠φ for each p ∈ π, A has an (?)-decomposition.展开更多
In this paper, we characterize some simple Ka-groups only by using the group order and largest element orders, where a simple K4-group is a simple group of order containing exactly four distinct primes.
In the paper we obtain two infinite classes of p-groups, calculate the orders of their automorphism groups and correct a mistake（perhaps misprinted） of Rodney James＇ paper in 1980.
In this paper by techniques of group we give some formulae for solving the general quadratic equations of two variables over a finite field,completely calculate and uniformly deal with the orders of automorphism group...In this paper by techniques of group we give some formulae for solving the general quadratic equations of two variables over a finite field,completely calculate and uniformly deal with the orders of automorphism groups of all p-groups of orders less than p6 under the P Hall's concept of isoclinism,also make a number of corrections for orders of automorphism groups offered for a mistake or fault before.展开更多
In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then ...In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then G is isomorphic to M if and only if the set of the orders of the maximal abelian subgoups of G is the same as that of M .展开更多
In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_...In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_z^y） is solvable.We also prove that group GL（n,Z_p）is solvable if and only if GL（n,Z_p） is solvable,and list the generators of groups GL（n,Z_p） and SL（n,Z_p）.At last,we prove that PSL（2,Z_p）（ p〉3） and PSL（n,Z_p） （ n〉3） are simple.展开更多
Jajcay's studies( 1993 ; 1994) on the automorphism groups of Cayley maps yielded a new product of groups, which he called, rotary product. Using this product, we define a hyperoperation ⊙ on the group Syme (G) , ...Jajcay's studies( 1993 ; 1994) on the automorphism groups of Cayley maps yielded a new product of groups, which he called, rotary product. Using this product, we define a hyperoperation ⊙ on the group Syme (G) , the stabilizer of the identity e ∈ G in the group Sym (G) . We prove that ( Syme (G) , ⊙) is a hypergroup and characterize the subhypergroups of this hypergroup.Finally, we show that the set of all subhypergroups of Syme ( G ) constitute a lattice under ordinary join and meet and that the minimal elements of order two of this lattice is a subgroup of Aut (G) .展开更多
This work deals with the power exponent 1rand 2r respectively of the maximal and second-maximal prime factors of the order of simple K4-group, and the classification for simple 4{5,7}K--group G (i.e. G can not be divi...This work deals with the power exponent 1rand 2r respectively of the maximal and second-maximal prime factors of the order of simple K4-group, and the classification for simple 4{5,7}K--group G (i.e. G can not be divided by 5 nor by 7 or ()Gp= 4 ), simple 5 -4K-group G (i.e. G can not divided by 5 and ()Gp=4) and simple 7-4K-group G (i.e. G can not divided by 7 and ()Gp= 4). It is derived that 1r =1, 2 and 4, and 2r is not greater than 4. All the simple 4K-groups with order 235,237abcdabcdpp and 2357abcd are obtained.展开更多
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.
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In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.
文摘Using a new symmetry group theory, the transformation groups and symmetries of the general Broer-Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.
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.
基金Climb-Up (Pan Deng) Project of Department of Science and Technology of China，国家自然科学基金，Doctoral Programme Foundation of Institution of Higher Education of China
文摘Canonical differential calculi are defined for finitely generated Abelian groups with involutions existing consistently. Two such the canonical calculi are presented. Fermionic representations for canonical calculi are defined based on quantized calculi. Fermionic representations for aforementioned two canonical calculi are searched out.
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.
文摘Paper considers the calculation of the values of Gibbs derivatives on finite Abelian groups. The calculation procedure is based upon the decision diagram representation of functions defined on finite Abelian groups. Approach permits processing of large functions.
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Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained： if G∈F, f（ ∞ ） include f（p）, f（p） ≠φ for each p∈π, and A has no nonzero F central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F central ZG-factors.
基金TheNationalNaturalScienceFoundationofChina (No .10 1710 74 )
文摘Let Fbe a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A a noetherian ZG-module with all irreducible ZG-factors being finite, G∈F, f(∞)f(p), f(p)≠ for each p∈π. The following conclutions are obtained: (1) if there exists a maximal submodule B of A such that A/B is F-central in G and B has no nonzero F-central ZG-factors, then A has an F-decomposition; (2) if there exists an irreducible F-central submodule B of A such that all ZG-composition factors of A/B are F-ecentric, then A has an F-decomposition.
文摘Let (?) be a formation locally defined by f(P), G ∈ (?) and A a ZG-module, where p ∈ π = { all primes and symbol ∞}. Then a p-main-factor U/V of G is said to be (?)-central in G if G/CG(U/V) ∈f(p). In this paper, we have proved that: let (?) be a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A an artinian ZG-module with all irreducible ZG-factors of A being finite; if G ∈ (?), f(∞) ≡ f(p) . f(p)≠φ for each p ∈ π, A has an (?)-decomposition.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1117136411271301)+2 种基金the Natural Science Foundation Project of CQ CSTC(Grant No.2014jcyjA00004)the Science and Technology Project of Chongqing Education Committee(Grant No.KJ1400520)the Foundation Project of Chongqing Normal University(Grant No.14XYY026)
文摘In this paper, we characterize some simple Ka-groups only by using the group order and largest element orders, where a simple K4-group is a simple group of order containing exactly four distinct primes.
基金Supported by NNSF of China(60574052)Supported by NSF(05001820)Supported by PST of Guangdong(2005B33301008)
文摘In the paper we obtain two infinite classes of p-groups, calculate the orders of their automorphism groups and correct a mistake（perhaps misprinted） of Rodney James＇ paper in 1980.
基金Supported by the National Natural Science Foundation of China(61074185)Supported by the Projection of Science and Technique for Guangdong Province(2009B030802044)+1 种基金Supported by the Projection of Production,Study and Investigation for Guangdong Province(2010B090301042)Supported by the Science and Study Foundation of Guangxi University(XB2100285)
文摘In this paper by techniques of group we give some formulae for solving the general quadratic equations of two variables over a finite field,completely calculate and uniformly deal with the orders of automorphism groups of all p-groups of orders less than p6 under the P Hall's concept of isoclinism,also make a number of corrections for orders of automorphism groups offered for a mistake or fault before.
文摘In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then G is isomorphic to M if and only if the set of the orders of the maximal abelian subgoups of G is the same as that of M .
文摘In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_z^y） is solvable.We also prove that group GL（n,Z_p）is solvable if and only if GL（n,Z_p） is solvable,and list the generators of groups GL（n,Z_p） and SL（n,Z_p）.At last,we prove that PSL（2,Z_p）（ p〉3） and PSL（n,Z_p） （ n〉3） are simple.
文摘Jajcay's studies( 1993 ; 1994) on the automorphism groups of Cayley maps yielded a new product of groups, which he called, rotary product. Using this product, we define a hyperoperation ⊙ on the group Syme (G) , the stabilizer of the identity e ∈ G in the group Sym (G) . We prove that ( Syme (G) , ⊙) is a hypergroup and characterize the subhypergroups of this hypergroup.Finally, we show that the set of all subhypergroups of Syme ( G ) constitute a lattice under ordinary join and meet and that the minimal elements of order two of this lattice is a subgroup of Aut (G) .
文摘This work deals with the power exponent 1rand 2r respectively of the maximal and second-maximal prime factors of the order of simple K4-group, and the classification for simple 4{5,7}K--group G (i.e. G can not be divided by 5 nor by 7 or ()Gp= 4 ), simple 5 -4K-group G (i.e. G can not divided by 5 and ()Gp=4) and simple 7-4K-group G (i.e. G can not divided by 7 and ()Gp= 4). It is derived that 1r =1, 2 and 4, and 2r is not greater than 4. All the simple 4K-groups with order 235,237abcdabcdpp and 2357abcd are obtained.
