Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are...Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.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
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
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
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
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
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 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
Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept： we say that H1 〈 H2〈.. 〈 Hk is a chain of nonnormal subgroups of G if each Hi G and if ｜Hi ： Hi-1｜ = p for i = 2,...Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept： we say that H1 〈 H2〈.. 〈 Hk is a chain of nonnormal subgroups of G if each Hi G and if ｜Hi ： Hi-1｜ = p for i = 2, 3,..., k. k is called the length of the chain, chn（G） denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn（G） ≤ 2 are completely classified up to isomorphism.展开更多
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
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
文摘Let F be a number field and p be a prime. In the successive approximation theorem, we prove that, for each integer n ≥ 1, finitely many candidates for the Galois group of the nth stage of the p-class tower over F are determined by abelian type invariants of p-class groups C1pE of unramified extensions E/F with degree [E : F] = pn-1. Illustrated by the most extensive numerical results available currently, the transfer kernels (TE, F) of the p-class extensions TE, F : C1pF → C1pE from F to unramified cyclic degree-p extensions E/F are shown to be capable of narrowing down the number of contestants significantly. By determining the isomorphism type of the maximal subgroups S G of all 3-groups G with coclass cc(G) = 1, and establishing a general theorem on the connection between the p-class towers of a number field F and of an unramified abelian p-extension E/F, we are able to provide a theoretical proof of the realization of certain 3-groups S with maximal class by 3-tower groups of dihedral fields E with degree 6, which could not be realized up to now.
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.
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.
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.
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.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 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.
文摘Assume that G is a finite non-Dedekind p-group. D. S. Passman introduced the following concept： we say that H1 〈 H2〈.. 〈 Hk is a chain of nonnormal subgroups of G if each Hi G and if ｜Hi ： Hi-1｜ = p for i = 2, 3,..., k. k is called the length of the chain, chn（G） denotes the maximum of the lengths of the chains of nonnormal subgroups of G. In this paper, finite 2-groups G with chn（G） ≤ 2 are completely classified up to isomorphism.
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 the National Natural Science Foundation of China(Nos.11771129,11301150,11601121)the Natural Science Foundation of Henan Province of China(No.162300410066)
文摘Let G be a finite p-group with a cyclic Frattini subgroup. In this paper, the automorphism group of G is determined.
基金Supported by NSFC (Nos.11701254,12061030)Education and Teaching Reform Project of Lingnan Normal University (No.LSJGYB1922)Key Subject Program of Lingnan Normal University (No.1171518004)。
