Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p &g...Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.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
Following Blackburn, Deaconescu and Mann, a group G is called an equilibrated group if for any subgroups H,K of G with HK = KH, either H≤NG(K) or K≤NG(H). Continuing their work and based on the classification of met...Following Blackburn, Deaconescu and Mann, a group G is called an equilibrated group if for any subgroups H,K of G with HK = KH, either H≤NG(K) or K≤NG(H). Continuing their work and based on the classification of metacyclic p-groups given by Newman and Xu, we give a complete classification of 2-generator equilibrated p-groups in this note.展开更多
In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples(k, l, β) ∈ N3 and all the subgroups of K. The results are applied...In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples(k, l, β) ∈ N3 and all the subgroups of K. The results are applied to compute some subgroups of K as well as to study the structure and the number of p-subgroups of K, where p is a fixed prime number. In addition, the author gets a factorization of K, and then studies the metacyclic p-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic p-group when p ≥ 3. A "reciprocity" relation on enumeration of subgroups of a metacyclic group is also given.展开更多
By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic nor...By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D3. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ5), by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K. The precise structure of the F5-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (UN:U0).?The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.展开更多
The Eisenstein Criterion is a sharp tool, i.e. a sufficient condition to judge a polynomial irreducible by using the coefficients of the polynomial. In particular, it plays an important role in studying totally ramifi...The Eisenstein Criterion is a sharp tool, i.e. a sufficient condition to judge a polynomial irreducible by using the coefficients of the polynomial. In particular, it plays an important role in studying totally ramified extensions of local fields. In this report, we shall generalize the Eisenstein Criterion for totally ramified extensions of local fields, and then we can get some results about extensions of local fields, that is to say, the conditions of tamely ramified extensions are cyclic or metacyclic.展开更多
Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.
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
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.
The author carries out a detailed study of p-blocks with normal metacyclic defect groups provided that p is odd and the defect groups are splitting and nonabelian.In particular,the author constructs all the irreducibl...The author carries out a detailed study of p-blocks with normal metacyclic defect groups provided that p is odd and the defect groups are splitting and nonabelian.In particular,the author constructs all the irreducible ordinary and modular characters in this type of blocks.As a by-product,k(B) and l(B) are obtained.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10671114)the Natural Science Foundation of Shanxi Province (Grant No. 2008012001)the Returned Abroad-Student Fund of Shanxi Province (Grant No. [2007]13-56)
文摘Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.
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 partially supported by the National Natural Science Foundation of China (Grant No. 10671114)the Natural Science Foundation of Shanxi Province (Grant No. 20051007)
文摘Following Blackburn, Deaconescu and Mann, a group G is called an equilibrated group if for any subgroups H,K of G with HK = KH, either H≤NG(K) or K≤NG(H). Continuing their work and based on the classification of metacyclic p-groups given by Newman and Xu, we give a complete classification of 2-generator equilibrated p-groups in this note.
基金the National Natural Science Foundation of China(No.11331006)。
文摘In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples(k, l, β) ∈ N3 and all the subgroups of K. The results are applied to compute some subgroups of K as well as to study the structure and the number of p-subgroups of K, where p is a fixed prime number. In addition, the author gets a factorization of K, and then studies the metacyclic p-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic p-group when p ≥ 3. A "reciprocity" relation on enumeration of subgroups of a metacyclic group is also given.
文摘By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D3. Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ5), by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K. The precise structure of the F5-vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (UN:U0).?The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.
文摘The Eisenstein Criterion is a sharp tool, i.e. a sufficient condition to judge a polynomial irreducible by using the coefficients of the polynomial. In particular, it plays an important role in studying totally ramified extensions of local fields. In this report, we shall generalize the Eisenstein Criterion for totally ramified extensions of local fields, and then we can get some results about extensions of local fields, that is to say, the conditions of tamely ramified extensions are cyclic or metacyclic.
基金NSFC (No.10671114)NSF of Shanxi Province (No.20051007)the Returned Overseas(student) Fund of Shanxi province (No.[2007]13-56)
文摘Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.
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.
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.
文摘The author carries out a detailed study of p-blocks with normal metacyclic defect groups provided that p is odd and the defect groups are splitting and nonabelian.In particular,the author constructs all the irreducible ordinary and modular characters in this type of blocks.As a by-product,k(B) and l(B) are obtained.
