Weil Representations and Some Nonreductive Dual Pairs in Symplectic and Unitary Groups

We study the restriction of the Weil representations of symplectic and unitary groups to subgroups that are the centralizers of certain elements, and show that these are multiplicity free. This work is along the line of the Howe philosophy for so-called dual pairs in the symplectic and unitary groups, but our dual pairs are not reductive. We also obtain an explicit formula for the character of the restriction to the centralizer of a regular unipotent element.     

Non-Normal Pairs of Non-Euclidean Crystallographic Groups

Let be a non-Euclidean crystallographic group. is said tobe non-maximal if there exists a non-Euclidean crystallographicgroup ' such that ' and the dimension of the Teichmüllerspace of equals the dimension of the Teichmüller spaceof '. The full list of such pairs of groups is computed in thecase when is non-normal in '. The corresponding problem forFuchsian groups was solved by Singerman. 2000 Mathematics SubjectClassification 20H10 (primary), 30F10 (secondary).     

有限群极大子群的θ-子群偶

N.P.Mukherjee和 P.Bhattacharya在“On theta pairs for a maximal sub-group”(Proc.Amer.Math.Soc,Vl09N3(1990))一文中定义了有限群的极大子群的θ-子群偶概念,研究了极大子群的极大θ-子群偶对群结构的影响,得到了一系列结果.本文在进一步探究θ-子群偶性质的基础上,对该文中一系列主要结果作出了本质性的改进,并给出了可解性、幂零性的一些新刻划.     

Theta Pairs and the Structure of Finite Groups

We introduce the notion of -pair of a proper subgroup of a finite group and give necessary and sufficient conditions for supersolvability and nilpotence of a finite group. As application, we give a proof to a recent open problem of Skiba in the unsolved problems in group theory.     

有限群的某些定理

本文给出了有限群的如下结果：１）成为可解群与超可解群的条件；２）推广了Ｉｔｏ定理，３）超可解且内幂零的群结构。     

The Carter Subgroups of Some Classical Groups

It is shown that a Carter subgroup of a non-soluble unitarygroup can only be the normalizer of a Sylow 2-subgroup. Thisresult, combined with previous results, implies that no finitesimple group can be a minimal counterexample to the conjugacyconjecture of Carter subgroups in a finite group. 2000 MathematicsSubject Classification 20D25, 20D05, 20E07.     

The Yagita Invariant of Some Symplectic Groups

If a group G of finite virtual cohomological dimension has p-periodic Farrell cohomology, then the Yagita invariant of this group equals its p-period. The group of symplectic 2(p + 1) × 2(p + 1) matrices over ${\mathbb{Z}}$ has elementary abelian p-subgroups of rank at least 2 and hence ${{\rm Sp}(2(p+1), \mathbb{Z})}$ and ${{\rm Sp}(2(p+1), \mathbb{Q})}$ do not have p-periodic Farrell cohomology. We compute the Yagita invariants of these groups for any odd prime p.     

Some Properties of Free Groups of Some Soluble Varieties of Groups

Let F be a free group, and let _n(F) be the nth term of the lowercentral series of F. It is proved that F/[_j(F), _i(F), _k(F)]and F/[_j(F), _i(F), _k(F), _l(F)] are torsion free and residuallynilpotent for certain values of i, j, k and i, j, k, l, respectively.In the process of proving this, it is proved that the analogousLie rings are torsion free.     

群的几个幂零性条件

如果G的所有子群都是次正规的,而且G满足下面条件之一,那么G是幂零群.(1)G有一个次正规列1△H△K△G,其中K/H是幂零群,H和G/K是有限生成的;(2)G有一个正规子群N使得,N在其子集的中心化子上满足极小条件,并且G/N是有限生成的.     

可换全序群的极大上下对

ＬｅｔＧｂｅａｎａｂｅｌｉａｎｏ ｇｒｏｕｐ (ｓｅｅ[1 ] )ｔｈｒｏｕｇｈｏｕｔａｎｄａ ,ｂ∈Ｇ .Ｉｆａ =0ｏｒ ｂ ａ ,ｔｈｅｎｗｅｓａｙｔｈａｔａｌｉｅｓａｂｏｖｅｂｏｒｂｌｉｅｓｂｅｌｏｗａ .ＬｅｔＡ ,ＢｂｅｔｗｏｓｕｂｇｒｏｕｐｓｏｆＧ .ＩｆｅａｃｈｅｌｅｍｅｎｔｏｆＡｌｉｅｓａｂｏｖｅａｌｌｅｌｅｍｅｎｔｓｏｆＢ ,ｔｈｅｎｗｅｓａｙｔｈａｔＡｌｉｅｓａｂｏｖｅＢｏｒＢｌｉｅｓｂｅｌｏｗＡ .ＩｆＡｉｓｍａｘｉｍａｌａｍｏｎｇｔｈｏｓｅｓｕｂｇｒｏｕｐｓｔｈａｔｌｉｅａｂｏｖｅＢ ,ｔｈｅｎｗｅ…     

几类非Abel群之增广商群的结构

记ZG为有限群G的整群环,△n(G)为增广理想△(G)的n次幂,Qn(G)=△"(G)/△n 1(G)为G的增广商群.本文考虑了二面体群D2tk(k 奇)和m次对称群Sm,证明了Qn(D2tk)为秩不超过2t 1的基本2-群以及Qn(Sm)≌Z2.     

Remarks on Gelfand Pairs Associated to Non-Type-I Solvable Lie Groups

LetSbe a connected and simply connected unimodular solvable Lie group andKa connected compact Lie group acting onSas automorphisms. We call the pair (K S) a Gelfand pair if the Banach ∗-algebraL¹_K(S) of allK-invariant integrable functions onSis a commutative algebra. In this paper we give a necessary and sufficient condition for the pair (K; S) to be a Gelfand pair using the representation theory of non-type-I solvable Lie groups. For a Gelfand pair (K; S) we realize all irreducibleK-spherical representations ofK?Sfrom irreducible unitary representations ofS.     

Some extremal problems for Pairs of functions with no common values

By the method of the moduli of families of curves one solves some extremal problems in the family of pairs {f₁, f₂} of functions f₁(z)=αz + ..., f₂(z)=βz^-1 + β⁰ +β¹z+..., with real coefficients, univalent, regular, resp. meromorphic in the circle Δ={∥z∥<1} and mapping onto nonoverlapping domains. As a special case the solution of a problem, posed by V. M. Miklyukov in Sib. Mat. Zh., Vol. 18, 1977, No. 5, pp. 1111–1124, is obtained. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 159–169, 1987.     

Test Rank for Some Free Polynilpotent Groups

We prove a theorem on possible test rank values for groups of the form . It is shown that test rank of a free polynilpotent group is equal to or , for any and every collection of classes. Moreover, for and .     

Some Combinatorial Results for Complex Reflection Groups

In this paper, we prove that a simple system for a subsystem Ψ of the complex root system Φ can always be chosen as a subset of the positive system Φ⁺of Φ. Furthermore, we show that a set of distinguished coset representatives can be found for every reflection subgroup of the complex reflection groups. The corresponding results for real crystallographic root systems and their reflection groups (i.e., Weyl groups) are well known (see [9]).     

Interpretable Groups, Stably Embedded Sets, and Vaughtian Pairs

The paper concerns sufficiently saturated structures M overa countable language with a unary predicate P. It is shown thatif P(M)is stably embedded and there are no Vaughtian pairs withrespect to P, then an infinite group is interpretable over M(in an infinitary sense of ‘interpretable’). Also,it is shown that if M is -categorical, f:DP is a 0-definablemap with finite fibres, and P(M) is stably embedded but D isnot, then some infinite group is first-order interpretable overM.     

The Normalizer Property for Integral Group Rings of Some Finite Nilpotent-by-Nilpotent Groups

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ?(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.     

On Some Noteworthy Pairs of Ideals in Mod-R

An additive functor $F \colon {\mathcal A}\to{\mathcal B}$ between preadditive categories $\mathcal A$ and $\mathcal B$ is said to be a local functor if, for every morphism $f\colon A\to A'$ in $\mathcal A$ , F(f) isomorphism in $\mathcal B$ implies f isomorphism in $\mathcal A$ . We show that there exist several pairs $(\mathcal I_1,\mathcal I_2)$ of ideals of $\mathcal A$ for which the canonical functor $\mathcal A\to\mathcal A/\mathcal I_1\times \mathcal A/\mathcal I_2$ is a local functor. In most of our examples, the category $\mathcal A$ is a full subcategory of the category Mod?-R of all right modules over a ring R. These pairs of ideals arise in a surprisingly natural way and enjoy several properties. Ideals are kernels of functors, and most of our examples of ideals are kernels of important and well studied functors. E.g., (1) the kernel Δ of the canonical functor P of Mod?-R into its spectral category Spec(Mod?-R), so that Δ is the ideal of all morphisms with an essential kernel; (2) the kernel Σ of the dual functor F of P, so that Σ is the ideal of all morphisms with a superfluous image; (3) the kernels Δ⁽¹⁾ and Σ₍₁₎ of the first derived functors P ⁽¹⁾ and F ₍₁₎ of P and F, respectively; (4) the kernels of suitable functors Hom and ? and their first derived functors ${\rm Ext}^1_R$ and ${\rm Tor}^R_1$ .     

若干家族p-群的自同构群的阶

本文确定了Rodney James文章中P．Hall isoclinism族第一家族到第十家族所有群的自同构群的阶,并纠正了Rodney James文章里的某些错误．