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
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
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
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
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.
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
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
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.
A subgroup of index p^k of a finite p-group G is called a k-maximal subgroup of G.Denote by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not ...A subgroup of index p^k of a finite p-group G is called a k-maximal subgroup of G.Denote by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G.In this paper,the authors classify the finite p-groups with δ_(d(G))(G) ≤ p^2 and δ_(d(G)-1)(G) = 0,respectively.展开更多
Assume G is a group of order p^n,where p is an odd prime.Let sk(G)denote the number of subgroups of order p^k of G.We give a criterion for a p-group to be with sk(G)≤p^4 for each integer k satisfying 1≤k≤n.Moreover...Assume G is a group of order p^n,where p is an odd prime.Let sk(G)denote the number of subgroups of order p^k of G.We give a criterion for a p-group to be with sk(G)≤p^4 for each integer k satisfying 1≤k≤n.Moreover,such p-groups are classified.展开更多
The classification of the reduced Abelian p-groups has been studied: Kaplansky proved that Ulm-Kaplansky invariants characterize the classification of countable groups; Kolettis extended this result to the direct sums...The classification of the reduced Abelian p-groups has been studied: Kaplansky proved that Ulm-Kaplansky invariants characterize the classification of countable groups; Kolettis extended this result to the direct sums of the countable groups; Parker and Walker further extended it to totally projective groups of length less than Ω_W; Hill proved that the greatest class of the p-groups which can be characterized by Ulm-Kaplansky invariants is the class of totally projective p-groups; Warfield have generalized this result to the simple presented modules in 1975.展开更多
A finite p-group P is called resistant if, for any finite group G having P as a Sylow p-group,the normalizer N_G(P) controls p-fusion in G. Let P be a central extension as 1→ Z_(p^m)→ P→ Z_p × · · ...A finite p-group P is called resistant if, for any finite group G having P as a Sylow p-group,the normalizer N_G(P) controls p-fusion in G. Let P be a central extension as 1→ Z_(p^m)→ P→ Z_p × · · · × Z_p→1,and |P'|≤p,m≥2. The purpose of this paper is to prove that P is resistant.展开更多
The Bogomolov multiplier B0 (G) of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality o...The Bogomolov multiplier B0 (G) of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether's problem. We show that if G is a central product of G1 and G2, regarding Ki ≤ Z(Gi),i = 1,2, and θ : G1 →G2 is a group homomorphism such that its restriction θ|K1 : K1 → K2 is an isomorphism, then the triviality of Bo(G1/K1), Bo(G1) and B0(G2) implies the triviality of Bo(G). We give a positive answer to Noether's problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement for the roots of unity).展开更多
Let G be a group and G' be its commutator subgroup. An automorphism α of a group G is called an IA-automorphism if x^-1α(x) ∈G' for each x∈G. The set of all IA-automorphisms of G is denoted by IA(G). A group...Let G be a group and G' be its commutator subgroup. An automorphism α of a group G is called an IA-automorphism if x^-1α(x) ∈G' for each x∈G. The set of all IA-automorphisms of G is denoted by IA(G). A group G is called semicomplete if and only if IA(G) = Inn(G), where Inn(G) is the inner automorphism group of G. In this paper we completely characterize semicomplete finite p-groups of class 2; we also classify all semicomplete finite p-groups of order p^n (n≤5), where p is an odd prime. This completes our work in 2011.展开更多
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
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.
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.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 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.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 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.
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 the National Natural Science Foundation of China(Nos.11371232,11371177)
文摘A subgroup of index p^k of a finite p-group G is called a k-maximal subgroup of G.Denote by d(G) the number of elements in a minimal generator-system of G and by δ_k(G) the number of k-maximal subgroups which do not contain the Frattini subgroup of G.In this paper,the authors classify the finite p-groups with δ_(d(G))(G) ≤ p^2 and δ_(d(G)-1)(G) = 0,respectively.
文摘Assume G is a group of order p^n,where p is an odd prime.Let sk(G)denote the number of subgroups of order p^k of G.We give a criterion for a p-group to be with sk(G)≤p^4 for each integer k satisfying 1≤k≤n.Moreover,such p-groups are classified.
文摘The classification of the reduced Abelian p-groups has been studied: Kaplansky proved that Ulm-Kaplansky invariants characterize the classification of countable groups; Kolettis extended this result to the direct sums of the countable groups; Parker and Walker further extended it to totally projective groups of length less than Ω_W; Hill proved that the greatest class of the p-groups which can be characterized by Ulm-Kaplansky invariants is the class of totally projective p-groups; Warfield have generalized this result to the simple presented modules in 1975.
基金Supported by NSFC(Grant Nos.11371154,11301150 and 11601121)Natural Science Foundation of Henan Province of China(Grant Nos.142300410134,162300410066)
文摘A finite p-group P is called resistant if, for any finite group G having P as a Sylow p-group,the normalizer N_G(P) controls p-fusion in G. Let P be a central extension as 1→ Z_(p^m)→ P→ Z_p × · · · × Z_p→1,and |P'|≤p,m≥2. The purpose of this paper is to prove that P is resistant.
基金Supported by Grant No.RD-08-82/03.02.2016 of Shumen University
文摘The Bogomolov multiplier B0 (G) of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier is an obstruction to Noether's problem. We show that if G is a central product of G1 and G2, regarding Ki ≤ Z(Gi),i = 1,2, and θ : G1 →G2 is a group homomorphism such that its restriction θ|K1 : K1 → K2 is an isomorphism, then the triviality of Bo(G1/K1), Bo(G1) and B0(G2) implies the triviality of Bo(G). We give a positive answer to Noether's problem for all 2-generator p-groups of nilpotency class 2, and for one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement for the roots of unity).
文摘Let G be a group and G' be its commutator subgroup. An automorphism α of a group G is called an IA-automorphism if x^-1α(x) ∈G' for each x∈G. The set of all IA-automorphisms of G is denoted by IA(G). A group G is called semicomplete if and only if IA(G) = Inn(G), where Inn(G) is the inner automorphism group of G. In this paper we completely characterize semicomplete finite p-groups of class 2; we also classify all semicomplete finite p-groups of order p^n (n≤5), where p is an odd prime. This completes our work in 2011.
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.