关于Q(4k~(2n)+1)的理想类群的循环子群

设ｄ,ａ,ｋ,ｎ是适合４ｋ２ｎ＋１＝ｄａ２,ｋ＞１,ｎ＞２,ｄ无平方因子的正整数;又设Ｃ（Ｋ）和ｈ（Ｋ）分别是实二次域Ｋ的理想类群和类数．本文证明了：当ａ＜０．５ｋ０．５６ｎ时,则ｈ（ｋ）＝０（ｍｏｄｎ）和Ｃ（Ｋ）必有ｎ阶循环子群．     

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = , or of twisted type G = ^m,m = 2,3, over a field K. Its root subgroups X_s, for all possible s G⁺, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U U⁹, g G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U U⁹ in G(K)has a p-multiple index. Put p() = max(r,r)/(s,s) | r,s . We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = and p() is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U of Uⁿ in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and (K,J) be a congruence subgroup of the Chevalley group (K) modulo a nilpotent ideal J. We examine an hypercentral series 1 Z₁ Z₂ ... Z_c-1 of the group U(K) (K,J). Theorem 2 shows that under an extra restriction on the quotient (J_t : J) of ideals, central series are related via Z_i = T_c-iC, 1 i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Z_pm and J = (p^d), 1 d < m, d| m.     

Definability in algebraically closed groups

Let K be an abstract class of groups such that a countable group U exists possessing the following properties: 1) an arbitrary finitely generated subgroup of U belongs to K; 2) an arbitrary finitely generated subgroup from K is imbedded in U; 3) a recursive representaion of the group U exists with a solvable word identity problem. Then for arbitrary n ≥ 1 there exists ??-equation Ψ_n(v₀...v_n?1) such that for an arbitrary algebraically closed group G and for arbitrary x₀...x_n?1 ε G Classes of finite free nilpotent groups satisfy the conditions of the theorem.     

典型群的增广商群

设G 为有限域K 上的一般线性群(特殊线性群、酉群、辛群及正交群), 记整群环ZG 的n 次增广理想为△ⁿ(G). 本文着重研究有限域上的典型群的增广商群Q_n(G) = △ⁿ(G)/△ⁿ⁺¹(G), 并刻画了这些连续商群的结构.     

Topological K-groups of two-dimensional local fields

A complete two-dimensional local field K of mixed characteristic with finite second residue field is considered. The existence of a completely ramified extension L of K such that L is a standard field is assumed. It is proved that the rank of the quotient U(1)K ₂ ^top K/T_K, where T_K is the closure of the torsion subgroup, is equal to the degree of the constant subfield of K over ℚ_p. I. B. Zhukov constructed a set of generators of this quotient in the case where K is a standard field. In this paper, two natural generalizations of this set are considered, and it is proved that one of them generates the entire group and the other generates its subgroup of finite index. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 206–221.     

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.     

典型域的调和函数论(Ⅲ)——斜对称方阵双曲空间的调和函数

<正> 3.1.斜对称方阵双曲空间的调和函数 命 Z 代表 n×n 斜对称方阵(?)(?)个复变数 z_(12),z_(13),…z_(1n),z_(23),…,(?)…,z_(n-1,n)空间的域我们引进运算子     

Complete p-Descent for Jacobians of Hermitian Curves

Let X be the Fermat curve of degree q+1 over the field k of q² elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.     

On the property <Emphasis Type="Italic">D</Emphasis>(2) and a common splitting field of two biquaternion algebras

Let F be a field of characteristic ≠ 2. We say that F possesses the property D(2) if for any quadratic extension L/F and any two binary quadratic forms over F having a common nonzero value over L, this value can be chosen in F. There exist examples of fields of characteristic 0 that do not satisfy the property D(2). However, as far as we know, it is still unknown whether there are such examples of positive characteristic and what is the minimal 2-cohomological dimension of fields for which the property D(2) does not hold. In this note it is shown that if k is a field of characteristic ≠ 2 such that |k*/k*²| ≥ 4, then for the field k(x) the property D(2) does not hold. Using this fact, we construct two biquaternion algebras over a field K = k(x)((t))((u)) such that their sum is a quaternion algebra, but they do not have a common biquadratic (i.e., a field of the kind , where a, b ∈ K*) splitting field. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 242–250.     

Bounded matrix groups

The principal result is that if G is any irreducible subgroup of GL(n, C) (C the complex field) such that |tr(7^)| is uniformly bounded for all A in G and some fixed n × n nonzero matrix T, then G is equivalent to a unitary group. Similar results are proved for certain associated representations of G.     

On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms} we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F_p. Let q be a power of p and let G(q) be the finite group of F_q-rational points of G. Let F be the Frobenius morphism such that G(q) = G^F. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) are in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).     

Bounded matrix groups

The principal result is that if G is any irreducible subgroup of GL(n, C) (C the complex field) such that |tr(7^)| is uniformly bounded for all A in G and some fixed n × n nonzero matrix T, then G is equivalent to a unitary group. Similar results are proved for certain associated representations of G.     

On Embeddings of AG(tn,k) into AG(t,K) with t≥2, n≥1

Let k and K be commutative fields with dim_kK=2n, n1 and char(k)2, 3. If k satisfies one of the following conditions: (1) k is a finite field and k contains a 3-th root of unity, or (2) K is a cyclic extension of k where k is not 3-closed, and k contains a 2^a3^b-th root of unity, where 6n=2^a3^bc and c is coprime to 2, 3, then there exists an embedding of the tn-dimensional affine space AG(tn,k) into the t-dimensional affine space AG(t,K), for all t2.     

A note on subgroups in a division ring that are left algebraic over a division subring

Let D be a division ring with center F and K a division subring of D. In this paper, we show that a non-central normal subgroup N of the multiplicative group $$D^*$$ is left algebraic over K if and only if so is D provided F is uncountable and contained in K. Also, if K is a field and the n-th derived subgroup $$D^{(n)}$$ of $$D^{*}$$ is left algebraic of bounded degree d over K, then $$\dim _FD\le d^2$$ .     

Frobenius Partitions in Extraspecial Groups

Let K be a field and G be the group of the upper unitriangular (n + 2) × ( n + 2) K-matrices with nonzero entries only in the first row and in the last column. Then G has a normal subgroup N with a complement H which are K-vector spaces respectively of dimensions n + 1 and n. In the present paper we show that the orbit of H under a group of automorphisms of G together with N, forms a partition of G, provided that there exists a commutative (possibly nonassociative) division algebra of dimension n + 1 over K. This algebra exists when K is a finite field.     

酉羣上的富理埃分析 Ⅰ.富理埃級数的Abel求和及Dirichlet核

<正> §1.1.引言 一个变数的Fourier分析,現在已經有了丰富的成果,不少問題得到了圓滿而完整的解决,在很多人的研究中达到了很深刻的地步.当然,一个变数的Fourier分析在数学的不少其他領域起着重要的作用.可是多个变数的Fourier分析情况就不完全是如此,在有些內容上是有些不完整之处的.至于更一般的在任意紧致羣上的Fourier分析,那末,众所周知的,重要的結果只是Peter-Weyl定理,它告訴我們說:紧致羣上的連續函数可以用