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.
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
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
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
In this paper, we determine the order of automorphism group of p-groups in the third family ( Φ 3) and the fourth family ( Φ 4) in [1], whose order is p^6(p≥3). Here p denotes an odd prime.
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
A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean (USC for short) if every element is uniquely the sum of an idempotent and a unit that commute...A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean (USC for short) if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring R and a group G such that RG is clean are given. It is also shown that if G is a locally finite group, then the group ring RG is USC if and only if R is USC, and G is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.展开更多
Using the metabelian property, regularity, p-commutativity and some properties of congruence, this paper gave the orders of automorphism groups of family Ф24, which are the groups of order p^6 determined by Rodney Ja...Using the metabelian property, regularity, p-commutativity and some properties of congruence, this paper gave the orders of automorphism groups of family Ф24, which are the groups of order p^6 determined by Rodney James, where p denotes an odd prime.展开更多
Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant ...Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.展开更多
Let Fq be a finite field of characteristic p (p ≠2) and V4 a four-dimensional Fq-vector space. In this paper, we mainly determine the structure of the transfer ideal for the ring of polynomials Fq[V4] under the actio...Let Fq be a finite field of characteristic p (p ≠2) and V4 a four-dimensional Fq-vector space. In this paper, we mainly determine the structure of the transfer ideal for the ring of polynomials Fq[V4] under the action of a nonmetacyclic p-group P in the modular case. We prove that the height of the transfer ideal is 1 using the fixed point sets of the elements of order p in P and that the transfer ideal is a principal ideal.展开更多
Recent examples of periodic bifurcations in descendant trees of finite p-groups with ?are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , ...Recent examples of periodic bifurcations in descendant trees of finite p-groups with ?are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , resp. (3,3), form periodic sequences in the descendant tree of the elementary Abelian root , resp. . The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.展开更多
The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K...The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.展开更多
Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their ...Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator qu-otient, the information content of each coclass subtree with metabelian main-line is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.展开更多
Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an ...Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.展开更多
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 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].
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.
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.
文摘In this paper, we determine the order of automorphism group of p-groups in the third family ( Φ 3) and the fourth family ( Φ 4) in [1], whose order is p^6(p≥3). Here p denotes an odd prime.
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A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean (USC for short) if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring R and a group G such that RG is clean are given. It is also shown that if G is a locally finite group, then the group ring RG is USC if and only if R is USC, and G is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.
基金The Science Research Foundation of Chongqing Municipal Education Commission of China(KJ050611)
文摘Using the metabelian property, regularity, p-commutativity and some properties of congruence, this paper gave the orders of automorphism groups of family Ф24, which are the groups of order p^6 determined by Rodney James, where p denotes an odd prime.
文摘Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.
文摘Let Fq be a finite field of characteristic p (p ≠2) and V4 a four-dimensional Fq-vector space. In this paper, we mainly determine the structure of the transfer ideal for the ring of polynomials Fq[V4] under the action of a nonmetacyclic p-group P in the modular case. We prove that the height of the transfer ideal is 1 using the fixed point sets of the elements of order p in P and that the transfer ideal is a principal ideal.
文摘Recent examples of periodic bifurcations in descendant trees of finite p-groups with ?are used to show that the possible p-class tower groups G of certain multiquadratic fields K with p- class group of type (2,2,2) , resp. (3,3), form periodic sequences in the descendant tree of the elementary Abelian root , resp. . The particular vertex of the periodic sequence which occurs as the p-class tower group G of an assigned field K is determined uniquely by the p-class number of a quadratic, resp. cubic, auxiliary field k, associated unambiguously to K. Consequently, the hard problem of identifying the p-class tower group G is reduced to an easy computation of low degree arithmetical invariants.
文摘The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.
文摘Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator qu-otient, the information content of each coclass subtree with metabelian main-line is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.
文摘Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.