E-Finitely generated groups

An abelian group G is said to be E-finitely generated (respectively, E-cyclic) if it is finitely generated (resp., cyclic) as a module over its endomorphism ring E. In this paper we refine a theorem of Reid [5] cUid apply the result to settle a question of some years standing concerning infinitely gen¬erated groups. We also determine the structure of torsion-free groups of finite rank for which Q ?_z is a progenerator over Q ?_z E     

On the splitting of groups of central group extensions

It is proved that if G and V are abelian groups, then the group Ext(G, V) of all abelian extensions of the group V with the aid of the group G is extracted by the direct component in the group Opext(G, V) of all central extensions of V with the aid of G.Translated from Matematicheskie Zametki, Vol. 6, No. 2, pp. 197–200, August, 1969.     

非极大交换子群皆正规的有限群

设H是有限群G的一个交换子群.如果H在G中的中心化子正是它本身,则称H为G的极大交换子群.本文主要研究每一非极大交换子群都正规的有限群的结构,对这类有限群给出其完全分类.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Extreme pairs for finite groups

Let G be a finite group with.O₂(G) = 1. If V is a faithful GF(2)G-module, then 𝔓 (G,V) is the set of elementary abelian 2-subgroups A of G with m(A) ? m(V/C_V(A)). A pair G,V is an extreme pair if G = < 𝔓 (G,V)>. Such pairs often appears in analysis of 2-local subgroups of characteristic 2-type groups. The following theorem is the main result of this paper.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

On commutative group algebras of mixed groups

Suppose F is a perfect field of characteristic p 0 and G is an abelian group whose torsion subgroup Gt is p-primary. If Gt is totally projective of countable length, it is shown that G is a direct factor of the group of normalized units V(G) of the group algebra F(G) and that V(G)/G is a totally projective p-group. The proof of this result is based on a new characterization of the class of totally projective p-groups of countable length. Li addition, the same result regarding V(G) is obtained if G has countable torsion-free rank and Gt is totally projective of length less than ω₁ + ω₀ . Finally, these results are applied to the question of whether the existence of an F-i pomorphism F(G) ? F(H) for some group H implies that G?H.     

Compact nilpotent groups

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π₂X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T _π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T _π-compact provided G/T _πG is π-complete, extending the abelian group result in a natural way.     

剩余有限minimax 可解群的几乎正则自同构

设G 是一个剩余有限的minimax 可解群, α 是G 的几乎正则自同构, 则G/[G, α] 是有限群, 并且(1) 当α^p = 1 时, G 有一个指数有限的幂零群其幂零类不超过h(p), 其中h(p) 是只与素数p 有关的函数.(2) 当α² = 1 时, G 有一个指数有限的Abel 特征子群且[G, α]′ 是有限群.关键词剩余有限minimax 可解群几乎正则自同构     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

Abelian groups as autocommutator groups

In this article, we establish that every finite abelian group is isomorphic to the autocommutator subgroup of some finite abelian group. Received: 21 September 2007     

有限Abel群的一个特征

本文证明了有限群Ｇ是Ａｂｅｌ群当且仅当Ｇ_ｒ满足下列条件：（Ⅰ） Ｇ有一个幂自同构 ａ使得 ＣＧ（ａ）是一个初等 ＡｂｅｌＺ一群．（Ⅱ）Ｇ没有子群与２－群＜ａ,ｂ｜ａ~２~ｎ＝ｂ~２~ｍ＝１,ａ~ｂ＝ａ~（１＋２）~（ｎ－１）＞同构,其中ｎ≥３,ｎ≥ｍ．利用该结果,作者还证明若有限群Ｇ有一个幂自同构ａ使得Ｃ_Ｇ（ａ）是一个初等Ａｂｅｌ２－群,则Ｇ是幂零群     

Characterization of infinite Chernikov groups

It is shown that an infinite locally finite group is a Chernikov group if and only if its Cartesian square G × G contains a subgroup T of finite index such that Aut T possesses the four group V as subgroup with Chernikov centralizer C_T(v) and the centralizers of involutions in V are weakly isolated in T.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 674–677, May, 1990.     

Homotopy equivalent group representations and Picard groups op the burnside ring and the character ring

We are concerned with the homotopy theory of group representations and its relation to character theory and the theory of the Burnside ring. We combine the methods of tom Dieck — Petrie [4] and torn Dieck [3] to show that the canonical map from the J-group jO(G), a subquotient of the representation ring RO(G), into the Picard group of the rational representation ring is injective for p-groups G. Moreover we compute the order of the cokernel of this map. We show that the Picard group of the rational representation ring is a direct summand in the Picard group of the Burnside ring. Finally we compute the Picard groups if G is abelian and indicate a computation for general G.     

具有阿贝尔Sylow 2-子群的有限群的整群环的正规化子性质

Mazur猜想:具有阿贝尔Sylow 2-子群的有限群有正规化子性质.设G是一个有限群,N是G的一个正规子群且Z(G/N)仅有平凡单位,本文建立了由Z(G/N)中单位诱导的G的自同构与N的Coleman自同构之间的联系,在此基础上证明了若G是一个具有阿贝尔Sylow 2-子群的有限群且Z(G/F*(G))仅有平凡单位,则Mazur猜想对G成立.     

On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

阿贝尔群被超-(循环或有限)群的可裂扩张(Ⅰ)

设F是由f(p)所局部定义的可解群系,G∈F,A是ZG-模.我们称A的一个p-主因子U/V在G中是F-中心的,如果G/CG(U/V)∈f(p).否则称U/V在G中是非中心的.本文证明了:设G是超-(有限或循环)的局部可解群,A是Artinian ZG-模且所有的不可约ZG-因子都是有限的;F为由f(p)所局部定义的局部可解群系,且对任意的p∈π,f(p)≠φ,f(∞) f(p).如果G∈F,且A的所有不可约ZG-因子在G中均是F-非中心的,则A被G的扩张在A上共轭可裂..     

A New Characterization of Simple K3-groups

ｔｈＮＮＴＷA New Characterization of Simple K_3-groups@王临红&&     

Finite groups with trivial Frattini subgroup

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.     

Fixed-point-free 2-finite automorphism groups

A fixed-point-free group G of automorphisms of an abelian group is shown to be locally finite if any two elements of G generate a finite subgroup.