Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

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.     

Some subgroups of semilinearly ordered groups

Let G be a semilinearly ordered group with a positive cone P. Denote byn(G) the greatest convex directed normal subgroup of G, byo(G) the greatest convex right-ordered subgroup of G, and byr(G) a set of all elements x of G such that x and x⁻¹ are comparable with any element of P^± (the collection of all group elements comparable with an identity element). Previously. it was proved thatr(G) is a convex right-ordered subgroup of G. andn(G) ⊆r(G) ⊆o(G). Here, we establish a new property ofr(G). and show that the inequalities in the given system of inclusions are, generally, strict. Supported by RFFR grant No. 99-01-00156. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 465–479, July–August, 2000.     

On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

Expanding curves in T1(Hn) under geodesic flow and equidistribution in homogeneous spaces

Let H = SO(n, 1) and A = {a(t): t ∈ R} be a maximal R-split Cartan subgroup of H. Let G be a Lie group containing H and Γ be a lattice of G. Let φ = gΓ ∈ G/Γ be a point of G/Γ such that its H-orbit Hx is dense in G/Γ. Let φ: I = [a, b] → H be an analytic curve. Then φ(I)x gives an analytic curve in G/Γ. In this article, we will prove the following result: if φ(I) satisfies some explicit geometric condition, then a(t)φ(I)x tends to be equidistributed in G/Γ as t → ∞. It answers the first question asked by Shah in [Sha09c] and generalizes the main result of that paper.     

Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

Equivariant holomorphic Hermitian principal bundles over a generalized flag manifold

Let G be a simple and simply connected complex linear algebraic group. Fix a maximal compact subgroup K(G) ì G{K(G) \subset G}, and let P be a parabolic subgroup of G. Let H be any connected reductive complex linear algebraic group. We classify the K(G)-equivariant holomorphic Hermitian principal H-bundles over G/P.     

A Note on Projective and Flat Dimensions and the Bieri-Neumann-Strebel-Renz Σ-Invariants

Let G be a finitely generated group, and A a ?[G]-module of flat dimension n such that the homological invariant Σ ⁿ (G, A) is not empty. We show that A has projective dimension n as a ?[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σ ⁿ (G, ?) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ^∞(H, ?) ≠ ?.     

Finite isometry groups of 4-manifolds with positive sectional curvature

Let M be an oriented compact Riemannian 4-manifold with positive sectional curvature. Let G be a finite subgroup of the isometry group of M. We prove that, if G is a finite group of order , then

On the Edge Connectivity of Direct Products with Dense Graphs

Let ${\kappa^{\prime}(G)}$ be the edge connectivity of G and G × H the direct product of G and H. Let H be any graph with minimal degree ${\delta(H)>|V(H)|/2}$ . We prove that for any graph ${G, \kappa^{\prime}(G\times H)=\textup{min}\{2\kappa^{\prime}(G)|E(H)|,\delta(G)\delta(H)\}}$ . In addition, the structure of minimum edge cuts is described. As an application, we present a necessary and sufficient condition for G × K _n (n ≥ 3) to be super edge connected.     

Characterization of 3 D 4(q)

The author defined the concept order components in [2] and gave a new characterization of sporadic simple groups by their order components in [7]. Afterwards the following groups were characterized by the author: G₂(q), q = 0 (mod 3)^[8]; E₈(q)^[9]; Suzuki-Ree groups^[10]; PSL₂(q)^[11]. Here the author will continue such kind of characterization and prove that:Theorem 1. Let G be a finite group, M = ³D₄(q). If G and M has the same order components, then G M.And the following theorems follows from Theorem 1.Theorem 2. (Thompsons Conjecture) Let G be a finite group, Z(G) = 1,M = ³D₄(q). If N(G) = N(M), then G M. (ref. [6])Theorem 3. (Wujie Shi) Let G be a finite group, M = ³D₄(q). If|G| = |M|, _e(G) = _e(M), then G M. (ref. [15])All notations are the same as in [2]. According to the classification theorem of finite simple groups, [12] and [13], we can list the order components of finite simple groups with nonconnected prime graphs in Tables 1-4 (ref. [5]).American Mathematics Society Classification 20D05 20D60The author is indebted to Fred and Barbara Kort Sino-Israel Postdoctoral Programme for supporting my post-doctoral position (1999.10-2000.10) at Bar-Ilan University, also to Emmy Noether Mathematics Institute and NSFC for partially financial support.     

The Hodge structure of the coloring complex of a hypergraph

Let G be a simple graph with n vertices. The coloring complex Δ(G) was defined by Steingrímsson, and the homology of Δ(G) was shown to be nonzero only in dimension n−3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group H_n−3(Δ(G)) where the dimension of the jth component in the decomposition, , equals the absolute value of the coefficient of λ^j in the chromatic polynomial of G, χ_G(λ).Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, Δ(H), and show that the coefficient of λ^j in χ_H(λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of Δ(H). We also examine conditions on a hypergraph, H, for which its Hodge subcomplexes are Cohen–Macaulay, and thus where the absolute value of the coefficient of λ^j in χ_H(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Δ(H). We also note that the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of jth term in the associated chromatic polynomial.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

The asymptotic σ-algebra of a recurrent random walk on a locally compact group

Let μ be a probability measure on a locally compact second countable groupG defining a recurrent (but not necessarily Harris) random walk. Denote byG ^∞ the space of paths and byB ^(a)the asymptotic σ-algebra. Let the starting measure be equivalent to the Haar measure and writeQ for the corresponding Markov measure onG ^∞. We prove thatL ^∞(G^∞, B^(a), Q) is in a canonical way isomorphic toL ^∞(G/N) whereN is the smallest closed normal subgroup ofG such that μ(zN)=1 for somez∈G. The groupG/N is either a finite cyclic group with generatorzN or a compact abelian group having the cyclic group as a dense subgroup. As a corollary we obtain that the set of all φ∈L ¹(G) such that coincides with the kernel of the canonical mapping ofL ¹(G)ontoL ¹(G/N). In particular, when μ is aperiodic, i.e.,G=N, then the random walk is mixing: for every φ∈L ¹(G) with ∝ φ=0.