Many of people have tried to obtain the structure of Baer-invariant of groups exactly. Recently, B. Mashayekhy and M. Parvizi determine Baer-invariant of finitely generated Abelian groups. Also, it is done for some no...Many of people have tried to obtain the structure of Baer-invariant of groups exactly. Recently, B. Mashayekhy and M. Parvizi determine Baer-invariant of finitely generated Abelian groups. Also, it is done for some non-Abelian groups, such the dihedral and the quaternion groups directly, and sometimes with the softwares of Gap and Magma. But nobody works on non finitely generated Abelian groups. In 1979, M.R.R. Moghaddarn showed that the structure of Baer-invariant of group commutes with the direct limit of a directed system, in some sense. The authors have used these results and proved that the Baer-invariant of C is always trivial and also Baer-invariant of Abelian groups Q/z and Z （p∞）, with respect to the varieties of outer commutators and so polynilpotent, nilpotent are trivial. One can see immediately that the covering groups of these groups are themselves. Then after computing the Baer-invariant of Zn with respect to Burnside variety, we have concluded for Q/z and Z（P∞） Burnside variety. In the future, they try to survey the commutativity of the Baer-invariant variety with the other useful varieties in order to attain similar results for another non finitely generated Abelian groups.展开更多
In this paper, our purpose is to make the results about π Frattini subgroup more accurate, and to extend Gaschütz Theorem about nilpotency to π locally defined formation. We come to Theorem L...In this paper, our purpose is to make the results about π Frattini subgroup more accurate, and to extend Gaschütz Theorem about nilpotency to π locally defined formation. We come to Theorem Let G be a finite group, H a subnormal subgroup of G. If H/H∩Φ(G)O π′ (G)∈F, then H∈F π, where F π is π solvable π locally defined formation.展开更多
In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for...In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for non-chief factor,there is no study up to now.The purpose of this paper is to build the theory.Let A be a subgroup of a finite group G and Σ:G0≤G1≤…≤Gn some subgroup series of G.Suppose that for each pair(K,H) such that K is a maximal subgroup of H and G i 1 K < H G i for some i,either A ∩ H = A ∩ K or AH = AK.Then we say that A is Σ-embedded in G.In this paper,we study the finite groups with given systems of Σ-embedded subgroups.The basic properties of Σ-embedded subgroups are established and some new characterizations of some classes of finite groups are given and some known results are generalized.展开更多
Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for etale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same ...Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for etale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.展开更多
文摘Many of people have tried to obtain the structure of Baer-invariant of groups exactly. Recently, B. Mashayekhy and M. Parvizi determine Baer-invariant of finitely generated Abelian groups. Also, it is done for some non-Abelian groups, such the dihedral and the quaternion groups directly, and sometimes with the softwares of Gap and Magma. But nobody works on non finitely generated Abelian groups. In 1979, M.R.R. Moghaddarn showed that the structure of Baer-invariant of group commutes with the direct limit of a directed system, in some sense. The authors have used these results and proved that the Baer-invariant of C is always trivial and also Baer-invariant of Abelian groups Q/z and Z （p∞）, with respect to the varieties of outer commutators and so polynilpotent, nilpotent are trivial. One can see immediately that the covering groups of these groups are themselves. Then after computing the Baer-invariant of Zn with respect to Burnside variety, we have concluded for Q/z and Z（P∞） Burnside variety. In the future, they try to survey the commutativity of the Baer-invariant variety with the other useful varieties in order to attain similar results for another non finitely generated Abelian groups.
文摘In this paper, our purpose is to make the results about π Frattini subgroup more accurate, and to extend Gaschütz Theorem about nilpotency to π locally defined formation. We come to Theorem Let G be a finite group, H a subnormal subgroup of G. If H/H∩Φ(G)O π′ (G)∈F, then H∈F π, where F π is π solvable π locally defined formation.
基金supported by National Natural Science Foundation of China (Grant No.11071229)Chinese Academy of Sciences Visiting Professorship for Senior International Scientists (Grant No.2010T2J12)
文摘In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for non-chief factor,there is no study up to now.The purpose of this paper is to build the theory.Let A be a subgroup of a finite group G and Σ:G0≤G1≤…≤Gn some subgroup series of G.Suppose that for each pair(K,H) such that K is a maximal subgroup of H and G i 1 K < H G i for some i,either A ∩ H = A ∩ K or AH = AK.Then we say that A is Σ-embedded in G.In this paper,we study the finite groups with given systems of Σ-embedded subgroups.The basic properties of Σ-embedded subgroups are established and some new characterizations of some classes of finite groups are given and some known results are generalized.
基金supported by the National Science Foundation(No.0900985)the National Security Agency(No.H98230-13-1-0209)+1 种基金the National Science Foundation(No.DMS-0757722)the Simons Foundation collaboration grant
文摘Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for etale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.
