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 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.
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
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 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 and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote th...Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote the unitary subgroup of V(FG).If p is odd,then the order of V_(*)(FG)is|F|^((|G-1)/2).However,the case p=2 still is open.In this paper,the order of V*(FG)is computed when G is a nonabelian 2-group given by a central extension of the form 1→Z_(2^(n))×Z_(2^(m))→G→Z_(2)×…×Z_(2)→1 and G'≌Z_(2),n,m≥1.Furthermore,a conjecture is confirmed,i.e.,the order of V_(*)(FG)can be divisible by|F|^(1/2(|G|+|Ω1(G)|)-1),where Ω_(1)(G)={g∈G|g^(2)=1}.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
Some new techniques are given to prove that two sorts of Abelian groups cannot function as the full automorphism groups of the finite groups. With generality these techniques made a breakthrough in MacHale’s problem.
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 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.
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.
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 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.
基金supported by National Natural Science Foundation of China(Grant No.12171142)。
文摘Let p be a prime and F be a finite field of characteristic p.Suppose that FG is the group algebra of the finite p-group G over the field F.Let V(FG)denote the group of normalized units in FG and let V_(*)(FG)denote the unitary subgroup of V(FG).If p is odd,then the order of V_(*)(FG)is|F|^((|G-1)/2).However,the case p=2 still is open.In this paper,the order of V*(FG)is computed when G is a nonabelian 2-group given by a central extension of the form 1→Z_(2^(n))×Z_(2^(m))→G→Z_(2)×…×Z_(2)→1 and G'≌Z_(2),n,m≥1.Furthermore,a conjecture is confirmed,i.e.,the order of V_(*)(FG)can be divisible by|F|^(1/2(|G|+|Ω1(G)|)-1),where Ω_(1)(G)={g∈G|g^(2)=1}.
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Project supported by the National Natural Science Foundation of China.
文摘Some new techniques are given to prove that two sorts of Abelian groups cannot function as the full automorphism groups of the finite groups. With generality these techniques made a breakthrough in MacHale’s problem.
