有限可解(q)群与(s—q)群的一个注记

文[l].[2]分别研究了每个次正规子群为拟正规的有限群(即(q)群)以及每个次正规子群为s—q拟正规的有限群(即(s—q)群).本文利用广幂零群的概念对(q)群与(s—q)群给出了一个新的刻划,并得到内(s—q)群的完全分类。     

Some remarks on almost l-groups

The divisibility group of every Bézout domain is an abelian l-group. Conversely, Jaffard, Kaplansky, and Ohm proved that each abelian l-group can be obtained in this way, which generalizes Krull’s theorem for abelian linearly ordered groups. Dumitrescu, Lequain, Mott, and Zafrullah [3] proved that an integral domain is almost GCD if and only if its divisibility group is an almost l-group. Then they asked whether the Krull-Jaffard-Kaplansky-Ohm theorem on l-groups can be extended to the framework of almost l-groups, and asked under what conditions an almost l-group is lattice-ordered [3, Questions 1 and 2]. This note answers the two questions. Received: 29 April 2008     

An answer to Hirasaka and Muzychuk: Every p-Schur ring over Cp3 is Schurian

In Hirasaka and Muzychuk [An elementary abelian group of rank 4 is a CI-group, J. Combin. Theory Ser. A 94 (2) (2001) 339–362] the authors, in their analysis on Schur rings, pointed out that it is not known whether there exists a non-Schurian p-Schur ring over an elementary abelian p-group of rank 3. In this paper we prove that every p-Schur ring over an elementary abelian p-group of rank 3 is in fact Schurian.     

On noninner automorphisms of 2-generator finite p-groups

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper we give some necessary conditions for a possible counterexample G to this conjecture, in the case when G is a 2-generator finite p-group. Then we show that every 2-generator finite p-group with abelian Frattini subgroup has a noninner automorphism of order p.     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.     

Determinazione deiT 2-gruppi finiti

Summary A T-group is a group in which normality is transitive, a T₁-group is a group which is not a T-group but all of whose proper subgroups are T-groups, and a T₂-group is a group which is not a T- group or a T₁-group but all of whose proper subgroups are T-groups or T₁-groups. In this note we determine all the finite T₂-groups, and we show that all T₂-groups which are either soluble or 2-groups are finite. We study also the groups in which every proper soluble subgroup is a T-group or a T₁-group.

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. (C.N.R.).     

Groups with given properties of finite subgroups

Suppose that in every finite even order subgroup F of a periodic group G, the equality [u, x]² = 1 holds for any involution u of F and for an arbitrary element x of F. Then the subgroup I generated by all involutions in G is locally finite and is a 2-group. In addition, the normal closure of every subgroup of order 2 in G is commutative.     

子群的核平凡或正规闭包极大的有限p群

称有限p群G为ACT群,如果对每个交换子群H,其正规核HG=1或HG=H.又称p群G是CC群,如果对每个非正规交换子群H,有HG=1或HG在G中的指数为p.本文分类了ACT群和CC群.     

关于半完全环的K1群

本文研究了半完全环的K1群. 利用半局部环的K1群的已知结果和半完全环的构造, 证明了半完全环R的K1群是R的一些子环的K1群的直和与R的单位群的一个子群的商群.     

The trace class conjecture in the theory of automorphic forms. II

Let G be the group of real points of a reductive algebraic -group satisfying the assumptions made in [H, Chapter I] and let be an arithmetic subgroup of G. Let be the right regular representation of G on and denote by the restriction of to the discrete subspace. In this paper we prove that for every integrable rapidly decreasing function f on G, the operator is of the trace class. Submitted: March 1997, Revised version: September 1997     

关于CAP-子群的一点注记（英文）

对于有限群G的每一主因子H/K来说,若G的子群L满足LH=LK或者L∩H=L∩K,则称L是G的CAP-子群.本文通过假设G的每个非循环Sylow子群P有一个子群D使得1〈｜D｜〈｜P｜,且P的所有阶为｜D｜和2｜D｜（若P是非交换2-群且｜P∶D｜〉2）的子群H是G的CAP-子群,得到G为p-幂零群的一个结果.     

Permutation presentations of modules over finite groups

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.     

一类可解的（q）－群

有限群Ｇ叫（ｑ）－群，如果Ｇ中每个次正规子群均为拟正规子群，群Ｇ叫Ｅｑ－群，若Ｇ中每个子群在Ｇ中拟正规或自正规，有限群Ｇ叫内Ｅｑ－群，如果Ｇ本身不是Ｅｑ－群，但Ｇ的每个真子群是Ｅｑ－群，本文确定了Ｅｑ－群的结构与内Ｅｑ－群的分类．     

Dieudonné Completion and PT-Groups

We consider the classes of PT-groups, strong PT-groups, completion friendly groups, and Moscow groups introduced by Arhangel’skii for the study of the Dieudonné completion of topological groups. We show that every subgroup H of a Lindel?f P-group is a PT-group, and that H is a strong PT-group iff it is \mathbb R{\mathbb R}-factorizable. Assuming CH, we prove that every ω-narrow P-group is a PT-group. Several results regarding products of PT-groups and \mathbb R{\mathbb R}-factorizable groups are established as well. We prove that the product of a Lindel?f group and an arbitrary subgroup of a Lindel?f Σ-group is completion friendly, and the same conclusion is valid for the product of an \mathbb R{\mathbb R}-factorizable P-group with an almost metrizable group.     

Holomorphic sections of line bundles over (H,C)-groups

Using an analogue of the refined Chern class, we study holomorphic sections of line bundles over an (H,C)-group. As an application we get that every meromorphically separable (H,C)-group is quasi-abelian.     

关于π''''-次数的不可约特征标在π-可分群里的一个注记

这篇文章中,我们研究了一种有限π-可分群G,它的每一个非线性不可约特征标是π-次的,得到了G是π-群的一个等价条件,并且给出了一些群理论的特征.     

Zur Struktur Der Inversen Teilbarkeits-Halbgruppen

In this paper we study inverse Tbk-semigroups [cf.1,2.]. We shall show that every inverse Tbk-semigroup is a subdirect product of 1-groups with or without 0, and we shall give necessary and sufficient conditions for (S,·,) to be a direct product of an 1-group and a distributive lattice. The second main result is that a Tbk-semigroup can be embedded into an inverse Tbk-semigroup iff as = ab = bb a = b and acx = acy axc = ayc.Originally this paper was part of [2]. It was separated from [2] at the suggestion of the referee.

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Karl Dörge zum Gedächtnis     

Σ-Groups as Convergence Groups

A Σ-group is an abelian group in which certain infinite sums are postulated to exist and to satisfy axioms suggested by the properties of unconditional sums in abelian convergence groups. Two notions of convergence are considered: in terms of nets, and in terms of filters. It is proved that every Σ-group can be regarded as a net convergence group (of a particular type). An example is given to show that the same does not remain true if filters are used instead of nets. For the special class of ‘adic’ Σ-groups, however, it is proved that filters are adequate.     

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