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
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
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
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
A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified...A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified completely in this paper.As an application,a problem proposed by Berkovich is solved.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 we classify regular p-groups with type invariants (e, 1, 1, 1) for e ≥ 2 and (1, 1, 1, 1, 1). As a by-product, we give a new approach to the classification of groups of order p5, p ≥ 5 a prime.
A subgroup H of a finite group G is called a TI-subgroup if H ∩ H^x = 1 or H for all x ∈ G. In this paper, a complete classification for finite p-groups, in which all abelian subgroups are TI-subgroups, is given.
Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question i...Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question introduced by Berkovich.展开更多
We classify finite p-groups all of whose nonnormal subgroups have orders at most p3, p odd prime. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prim...We classify finite p-groups all of whose nonnormal subgroups have orders at most p3, p odd prime. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.展开更多
A finite group is inseparable, it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect product decompositions. This paper investigates two classes of finite inseparabl...A finite group is inseparable, it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect product decompositions. This paper investigates two classes of finite inseparable p-groups and, for p ≥ 3, establishes a necessary and sufficient condition for insep- arability.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
Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p^3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonme tacyclic of or...Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p^3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonme tacyclic of order p^3. In this paper, the P1-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P1-groups.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
Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its pro...Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in de(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified. Keywords Finite p-groups, normal subgroups, subgroup complement展开更多
A subgroup A of a p-group G is said to be soft in G if CG(A) = A and |NG(A)/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.
Finite p-groups whose subgroups of given order are isomorphic and minimal non-abelian are classified. In addition, two results on a chain condition of At-groups are improved.
Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Th...Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Then G is a central product of an extraspecial pkgroup E andζG.Let|E|=p(2n+1)k and|ζG|=p(m+1)k.Suppose that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤k.Let AutG’G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G’,let AutG/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the centerζG and let AutG/ζG,ζG/G’G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially onζG/G’.Then(ⅰ)The group extension 1→Aut G’→Aut G→Aut G’→1 is split.(ⅱ)AutG’G/AutG/ζG,ζG G≌G1×G2,where Sp(2n-2,Zpk)■H≤G1≤Sp(2n,Zpk),H is an extraspecial pk-group of order p(2n-1)k and(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)≤G2≤GL(m,Zpk)■Zpk(m).In particular,G1=Sp(2n-2,Zpk)■H if and only if l=k and r=0;G1=Sp(2n,Zpx)if and only if l≤r;G2=(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)if and only if r=k;G2=GL(m,Zpk)■Zpk((m))if and only if r=0.(ⅲ)AutG’G/Aut G/ζG,ζG/G’G≌G1×G3,where G1 is defined in(ⅱ);GL(ml,Zpk)■Zpk(m-1)≤G3≤GL(n,Zpk).In particular,G3=GL(m-1,Zpk)■Zpk(m-1)if and only if r=k;G3=GL(m,Zpk)if and only if r=0.(ⅳ)AntG/ζG,ζG/G’G≌AutG/ζG,ζG/G’G■Zpk(m),If m=0,then AntG/ζG,ζG/G’G=Inn G≌Zpk(2n);If m>0,then AntG/ζG,ζG/G’G≌Zpk(2nm)×Zpk-r(2n),and AutG/ζG,ζG G/Inn G≌Zpk((2n(m-1))×Zpk-r(2n).展开更多
The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.
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 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.
基金supported by the National Natural Science Foundation of China(nos.12171213,11771191,11771258).
文摘A finite p-group G is called an At-group if t is the minimal non-negative integer such that all subgroups of index pt of G are abelian.The finite p-groups G with H'=G'for all A2-subgroups H of G are classified completely in this paper.As an application,a problem proposed by Berkovich is solved.
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 Founda tion of China(Grant Nos.10371003&10471085)Natural Science Foundation of Beijing 1052005)+2 种基金Natural Science Foundation of Shanxi Province(Grant No.20051007)Key Project of Ministry of Education(Grant No.02023)The Returned Abroad-Student Found of Shanxi Province(Grant No.[2004]7).
文摘In this paper we classify regular p-groups with type invariants (e, 1, 1, 1) for e ≥ 2 and (1, 1, 1, 1, 1). As a by-product, we give a new approach to the classification of groups of order p5, p ≥ 5 a prime.
基金the Natural Science Foundation of China(10161001)the Natural Science Foundation of Guangxi of China+1 种基金the National Natural Science Foundation of Shanghai Education CommitteeSpecial Funds for Major Specialities of Shanghai Education Committee
文摘A subgroup H of a finite group G is called a TI-subgroup if H ∩ H^x = 1 or H for all x ∈ G. In this paper, a complete classification for finite p-groups, in which all abelian subgroups are TI-subgroups, is given.
基金supported by National Natural Science Foundation of China (Grant Nos.10571128,10871032)Natural Science Foundation of Jiangsu Province (Grant No.BK2008156)Suzhou City Senior Talent Supporting Project
文摘Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question introduced by Berkovich.
基金Acknowledgements The authors cordially thank the referees for detailed and valuable comments, which help them to improve the paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11371232, 11101252), the Natural Science Foundation of Shanxi Province (No. 2012011001, 2013011001), and Shanxi Scholarship Council of China (No. [201118).
文摘We classify finite p-groups all of whose nonnormal subgroups have orders at most p3, p odd prime. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.
文摘A finite group is inseparable, it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect product decompositions. This paper investigates two classes of finite inseparable p-groups and, for p ≥ 3, establishes a necessary and sufficient condition for insep- arability.
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Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p^3. Let P1-groups denote the p-groups all of whose minimal nonabelian subgroups are nonme tacyclic of order p^3. In this paper, the P1-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P1-groups.
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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11471198,11501045 and 11371232)
文摘Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in de(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified. Keywords Finite p-groups, normal subgroups, subgroup complement
基金supported by National Natural Science Foundation of China (Grant No.11071150)Natural Science Foundation of Shanxi Province (Grant No. 2008012001)The Returned Abroad-student Found of Shanxi Province (Grant No. [2007]13–56)
文摘A subgroup A of a p-group G is said to be soft in G if CG(A) = A and |NG(A)/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.
文摘Finite p-groups whose subgroups of given order are isomorphic and minimal non-abelian are classified. In addition, two results on a chain condition of At-groups are improved.
基金Supported by NSFC(Grant Nos.11601121,11771129)Natural Science Foundation of He’nan Province of China(Grant No.162300410066)。
文摘Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Then G is a central product of an extraspecial pkgroup E andζG.Let|E|=p(2n+1)k and|ζG|=p(m+1)k.Suppose that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤k.Let AutG’G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G’,let AutG/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the centerζG and let AutG/ζG,ζG/G’G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially onζG/G’.Then(ⅰ)The group extension 1→Aut G’→Aut G→Aut G’→1 is split.(ⅱ)AutG’G/AutG/ζG,ζG G≌G1×G2,where Sp(2n-2,Zpk)■H≤G1≤Sp(2n,Zpk),H is an extraspecial pk-group of order p(2n-1)k and(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)≤G2≤GL(m,Zpk)■Zpk(m).In particular,G1=Sp(2n-2,Zpk)■H if and only if l=k and r=0;G1=Sp(2n,Zpx)if and only if l≤r;G2=(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)if and only if r=k;G2=GL(m,Zpk)■Zpk((m))if and only if r=0.(ⅲ)AutG’G/Aut G/ζG,ζG/G’G≌G1×G3,where G1 is defined in(ⅱ);GL(ml,Zpk)■Zpk(m-1)≤G3≤GL(n,Zpk).In particular,G3=GL(m-1,Zpk)■Zpk(m-1)if and only if r=k;G3=GL(m,Zpk)if and only if r=0.(ⅳ)AntG/ζG,ζG/G’G≌AutG/ζG,ζG/G’G■Zpk(m),If m=0,then AntG/ζG,ζG/G’G=Inn G≌Zpk(2n);If m>0,then AntG/ζG,ζG/G’G≌Zpk(2nm)×Zpk-r(2n),and AutG/ζG,ζG G/Inn G≌Zpk((2n(m-1))×Zpk-r(2n).
基金Project supported by NSFC (11371124, 11301150) and the Natural Science Foundation of Henan Province of China (142300410134, 162300410066).
文摘The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.