First Alexandroff Decomposition Theorem for Topological Lattice Group Valued Measures

Let (X, F) be an Alexandroff space, let A(F) be the Boolean subalgebra of 2 ^X generated by F, let G be a Hausdorff commutative topological lattice group and let rba_F(A(F), G) denote the set of all order bounded F-inner regular finitely additive set functions from A(F) into G. Using some special properties of the elements of rba_F(A(F), G), we extend to this setting the first decomposition theorem of Alexandroff.     

On the relative character correspondences

Let π(G,A):Irr_A(G)→Irr(C_G(A)) be the Glauberman–Isaacs correspondence, where G and A are finite groups with coprime orders and A acts on G by automorphisms. Let B be a subgroup of A. In this setting, we give some new conditions for the fixed-point subgroups C_G(A) and C_G(B) such that χπ(G,A) is an irreducible constituent of the restriction of χπ(G,B) to C_G(A) for all χIrr_A(G).     

On Orbits of Automorphism Groups

If G is a finite group and if A is a group of automorphisms of G whose fixed point subgroup is C _G (A) then every subgroup F of C _G (A) acts on the set of orbits of A in G. The peculiarities of this action are used here to derive several results on the number of orbits of A in an economical manner.Original Russian Text Copyright © 2005 Deaconescu M. and Walls G. L.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 533–537, May–June, 2005.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

Group actions and orbits

Let A be a group acting via automorphisms on a group G, and let Ω be the set of orbits in this action. Then C = C _G (A) acts on Ω in a natural manner, and using this action, we deduce some divisibility information about |Ω|.     

Maximal subnear-rings of functions

For a group G, let M(G) denote the near-ring of functions on G. We characterize all maximal subnear-rings of M(G) and show that for many classes of groups, E(G), the near-ring generated by the semigroup, End(G) of G, is never maximal as a subnear-ring of M ₀ (G). Received: 25 April 2008     

Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T ^*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2(G,K) the maximal number of functionally independent functions from A\C. We prove that (G,K) is equal to the codimension (G,K) of maximal dimension orbits of the Borel subgroup BG ^C in the complex algebraic variety G ^C/K ^C. Moreover, if (G,K)=1, then all G-invariant Hamiltonian systems on T ^*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.     

On the Chow Ring of a Flag

Let G be a reductive linear algebraic group over an algebraically closed field K, let P? be a parabolic subgroup scheme of G containing a Borel subgroup B, and let P = P?_red ? P? be its reduced part. Then P is reduced, a variety, one of the well known classical parabolic subgroups. For char(K) = p > 3, a classification of the P?'s has been given in [W1]. The Chow ring of G/P only depends on the root system of G. Corresponding to the natural projection from G/P to G/P? there is a map of Chow rings from A(G/P?) to A(G/P). This map will be explicitly described here. Let P = B, and let p > 3. A formula for the multiplication of elements in A(G/P?) will be derived. We will prove that A(G/P?) ? A(G/P) (abstractly as rings) if and only if G/P ? G/P? as varieties, i. e., the Chow ring is sensitive to the thickening. Furthermore, in certain cases A(G/P?) is not any more generated by the elements corresponding to codimension one Schubert cells.     

Matrix Near-rings over Centralizer Near-rings

For a finite group G and a subgroup A of Aut(G), let M_A(G) denote the centralizer near-ring determined by A and G. The group G is an M_A(G)-module. Using the action of M_A(G) on G, one has the n × n generalized matrix near-ring Mat_n(M_A(G);G). The correspondence between the ideals of M_A(G) and those of Mat_n(M_A(G);G) is investigated. It is shown that if every ideal of M_A(G) is an annihilator ideal, then there is a bijection between the ideals of M_A(G) and those of Mat_n(M_A(G);G).1991 Mathematics Subject Classification: 16Y30     

On one class of modules that are close to Noetherian

We consider an R G-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that C _G (A) = 1, A/C _A (G) is not a Noetherian R-module, but the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

On Schur classes for modules over group rings

We consider the problem of coupling between a quotient module A/C _A (G) and a submodule A(ωRG), where G is a group, R is a ring, and A is an RG-module; C _A (G) can be considered as an analog of the center of the group, and the submodule A(ωRG) can be considered as an analog of the derived subgroup of the group. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1261–1268, September, 2007.     

Sum of Element Orders of Maximal Subgroups of the Symmetric Group

For a finite group G, let ψ(G) denote the sum of element orders of G. The aim of this article is to show that ψ(H) < ψ(A _n ) for every proper subgroup H of the symmetric group of degree n, which is different from the alternating group A _n .     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

The Extension of Möbius Groups

Let G be a non-elementary subgroup of SL(2,Г _n ) containing hyperbolic elements. We show that G is the extension of a subgroup of SL(2,C) if and only if that G is conjugate in SL(2,Г _n ) to a group G' with the following properties: (1) There are g ₀, h ? G', where g ₀ and h are hyperbolic, such that fix(g ₀) = {0,∞}, fix(h)∩fix(g ₀) =  and fix(h) ∩ C ≠ ; (2) tr(g) ? C for each g ? G'. As an application, we show that if G contains only hyperbolic elements and uniformly parabolic elements, then G is the extension of a subgroup of SL(2,C), which also yields the discreteness of G.     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

On Some Sets of Group Functions

Let G be a group, let A be an Abelian group, and let n be an integer such that n –1. In the paper, the sets _n (G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set _n (G,A) is reduced to that of the set n(G/O ^p (G),A _p ) for primes p dividing G/G. Here O ^p (G) stands for the p-coradical of the group G, A _p for the p-component of A, and G for the commutator subgroup of G.     

Linear groups with minimality condition for some infinite-dimensional subgroups

Let F be a field, let A be a vector space over F, and let GL(F, A) be the group of all automorphisms of the space A. If H is a subgroup of GL(F, A), then we set aug dim_F (H) = dim_F (A(ωFH)), where ωFH is the augmentation ideal of the group ring FH. The number aug dim_F (H) is called the augmentation dimension of the subgroup H. In the present paper, we study locally solvable linear groups with minimality condition for subgroups of infinite augmentation dimension. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1476–1489, November, 2005.     

On Finite n-Acentralizer Groups

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.