A p-group G is called a JC-group if the normal closure HG of every cyclic subgroup H satisfies｜G：HG｜≤p or｜HG：H｜≤p.In this paper, we classify the non-Dedekindian JC-groups for p 〉 2.
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.
A finite p-group G is called an LA-group if |G|||Aut(G)| when G is non-cyclic and |G|>p^2. This paper shows that a p-group of order p^n with an element of order p^(n-2) is an LA-group.
For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic de...For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.展开更多
In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A co...In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.展开更多
Let p be a prime number and f_2(G) be the number of factorizations G = AB of the group G, where A, B are subgroups of G. Let G be a class of finite p-groups as follows,G = a, b | a^(p^n)= b^(p^m)= 1, a^b= a^(p^(n-1)+1...Let p be a prime number and f_2(G) be the number of factorizations G = AB of the group G, where A, B are subgroups of G. Let G be a class of finite p-groups as follows,G = a, b | a^(p^n)= b^(p^m)= 1, a^b= a^(p^(n-1)+1), where n > m ≥ 1. In this article, the factorization number f_2(G) of G is computed, improving the results of Saeedi and Farrokhi in [5].展开更多
Assume that N, F and G are groups. If there exsits N, a normal subgroup of G such that N ≌ G and GIN ≌ F, then G is called a central extension of N by F. In this paper, the central extension of N by a minimal non-ab...Assume that N, F and G are groups. If there exsits N, a normal subgroup of G such that N ≌ G and GIN ≌ F, then G is called a central extension of N by F. In this paper, the central extension of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group of order p3. Together with our previous work, all central extensions of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group.展开更多
Let G be a group and A and B be subgroups of G.If G=AB,then G is said to be factorized by A and B.Let p be a prime number.The factorization numbers of a 2-generators abelian p-group and a modular p-group have been det...Let G be a group and A and B be subgroups of G.If G=AB,then G is said to be factorized by A and B.Let p be a prime number.The factorization numbers of a 2-generators abelian p-group and a modular p-group have been determined.Further,suppose that G is a finite p-group as follows G=<a,b|a^(p)^(n)=b^(p)^(m)=1,a^(b)=a^(p^(n-1)+1)>,where n≥2,m≥1.In this paper,the factorization number of G is computed completely,which is a generalization of the result of Saeedi and Farrokhi.展开更多
Let G be a finite group. A nonempty subset X of G is said to be noncommuting if xy≠yx for any x, y ∈ X with x≠y. If ｜X｜ ≥ ｜Y｜ for any other non-commuting set Y in G, then X is said to be a maximal non-commutin...Let G be a finite group. A nonempty subset X of G is said to be noncommuting if xy≠yx for any x, y ∈ X with x≠y. If ｜X｜ ≥ ｜Y｜ for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, we determine upper and lower bounds on the cardinality of a maximal non-commuting set in a finite p-group with derived subgroup of prime order.展开更多
Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,thi...Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.展开更多
Suppose that G is a finite p-group.If all subgroups of index p^(t)of G are abelian and at least one subgroup of index p^(t−1)of G is not abelian,then G is called an A_(t)-group.We useA0-group to denote an abelian grou...Suppose that G is a finite p-group.If all subgroups of index p^(t)of G are abelian and at least one subgroup of index p^(t−1)of G is not abelian,then G is called an A_(t)-group.We useA0-group to denote an abelian group.From the definition,we know every finite non-abelian p-group can be regarded as an A_(t)-group for some positive integer t.A_(1)-groups and A_(2)-groups have been classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,we determine the Frattini subgroup,the derived subgroup and the center of every A_(3)-group,and give the number of A_(1)-subgroups and the triple(μ_(0),μ_(1),μ_(2))of every A_(3)-group,whereμi denotes the number of A_(i)-subgroups of index p of A_(3)-groups.展开更多
In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in w...In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.展开更多
Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^M（G） and p^m（G） to denote the maximum and minimum of the orders of the non-normal subgroups of G, respec...Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^M（G） and p^m（G） to denote the maximum and minimum of the orders of the non-normal subgroups of G, respectively. In this paper, we classify groups G such that M（G） 〈 2m（G） ^- 1. As a by-product, we also classify p-groups whose orders of non-normal subgroups are p^k and p^k＋1.展开更多
For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.Thi...For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.展开更多
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 National Natural Science Foundation of China(Grant Nos.1152611411601245)+3 种基金the Natural Science Foundation of Guangdong Province(Grant No.2015A030313791)the Innovative Team Project of Guangdong Province(CHINA)(Grant No.2014KTSCX196)Guangdong Province Innovation Talent Project for Youths(Grant No.2015KQNCX107)the Appropriative Researching Fund for Doctors,Guangdong University of Education(Grant No.2013ARF07)
文摘A p-group G is called a JC-group if the normal closure HG of every cyclic subgroup H satisfies｜G：HG｜≤p or｜HG：H｜≤p.In this paper, we classify the non-Dedekindian JC-groups for p 〉 2.
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.
文摘A finite p-group G is called an LA-group if |G|||Aut(G)| when G is non-cyclic and |G|>p^2. This paper shows that a p-group of order p^n with an element of order p^(n-2) is an LA-group.
基金Supported by the NSF of China(11171194)by the NSF of Shanxi Province(2012011001-1)
文摘For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.
文摘In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.
基金Supported by National Natural Science Foundation of China(11601121)Henan Provincial Natural Science Foundation of China(162300410066)
文摘Let p be a prime number and f_2(G) be the number of factorizations G = AB of the group G, where A, B are subgroups of G. Let G be a class of finite p-groups as follows,G = a, b | a^(p^n)= b^(p^m)= 1, a^b= a^(p^(n-1)+1), where n > m ≥ 1. In this article, the factorization number f_2(G) of G is computed, improving the results of Saeedi and Farrokhi in [5].
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137123211471198)
文摘Assume that N, F and G are groups. If there exsits N, a normal subgroup of G such that N ≌ G and GIN ≌ F, then G is called a central extension of N by F. In this paper, the central extension of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group of order p3. Together with our previous work, all central extensions of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group.
基金Supported by National Natural Science Foundation of China(Grant No.11601121,12171142).
文摘Let G be a group and A and B be subgroups of G.If G=AB,then G is said to be factorized by A and B.Let p be a prime number.The factorization numbers of a 2-generators abelian p-group and a modular p-group have been determined.Further,suppose that G is a finite p-group as follows G=<a,b|a^(p)^(n)=b^(p)^(m)=1,a^(b)=a^(p^(n-1)+1)>,where n≥2,m≥1.In this paper,the factorization number of G is computed completely,which is a generalization of the result of Saeedi and Farrokhi.
基金The NSF(11301150,11371124)of Chinathe NSF(142300410134)of Henan ProvincePlan for Scientific Innovation Talent(11CXRC19)of Henan University of Technology
文摘Let G be a finite group. A nonempty subset X of G is said to be noncommuting if xy≠yx for any x, y ∈ X with x≠y. If ｜X｜ ≥ ｜Y｜ for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, we determine upper and lower bounds on the cardinality of a maximal non-commuting set in a finite p-group with derived subgroup of prime order.
文摘Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.
基金supported by the National Natural Science Foundation of China(Nos.11371232,11101252)the Shanxi Provincial Natural Science Foundation of China(No.2013011001)the Fundamental Research Funds for the Central Universities(No.BUPT2013RC0901)
文摘The groups as mentioned in the title are classified up to isomorphism. This is an answer to a question proposed by Berkovich and Janko.
基金This work was supported by NSFC(Nos.11371232,11471198)by NSF of Shanxi Province(No.2013011001).
文摘Suppose that G is a finite p-group.If all subgroups of index p^(t)of G are abelian and at least one subgroup of index p^(t−1)of G is not abelian,then G is called an A_(t)-group.We useA0-group to denote an abelian group.From the definition,we know every finite non-abelian p-group can be regarded as an A_(t)-group for some positive integer t.A_(1)-groups and A_(2)-groups have been classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,we determine the Frattini subgroup,the derived subgroup and the center of every A_(3)-group,and give the number of A_(1)-subgroups and the triple(μ_(0),μ_(1),μ_(2))of every A_(3)-group,whereμi denotes the number of A_(i)-subgroups of index p of A_(3)-groups.
基金supported by the National Natural Science Foundation of China(No.10671114)the ShanxiProvincial Natural Science Foundation of China(No.2008012001)the Returned Abroad-StudentFund of Shanxi Province(No.[2007]13-56)
文摘In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11471198, 11771258）.
文摘Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^M（G） and p^m（G） to denote the maximum and minimum of the orders of the non-normal subgroups of G, respectively. In this paper, we classify groups G such that M（G） 〈 2m（G） ^- 1. As a by-product, we also classify p-groups whose orders of non-normal subgroups are p^k and p^k＋1.
基金supported by National Natural Science Foundation of China(Grant Nos.11771258 and 11471198)。
文摘For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.
