Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionI ⁿ⁺¹ (F) ∩I ⁿ⁺¹ (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I ³ (F)) of F are identified whenR and S are arbitrary subgroups ofF.     

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.     

Subgroups Determined by Certain Products of Augmentation Ideals

Let G be a group, ZG the integral group ring of G, and I(G) its augmentation ideal. Let H be a subgroup of G. It is proved that the subgroup of G determined by the product I(H)I(G)I(H) equals 3(H), i.e., the third term in the lower central series of H. Also, the subgroup determined by I(H)I(G)Iⁿ(H) (resp., Iⁿ(H)I(G)I(H)) for n > 1 equals D_n+2(H), the (n + 2)th dimension subgroup of H.Supported by the National Board for Higher Mathematics, India.1991 Mathematics Subject Classification: 20C05, 20C07     

Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

Isomorphisms ofp-adic group rings

SupposeP is the ring ofp-adic integers,G is a finite group of orderp ⁿ , andPG is the group ring ofG overP. IfV _p (G) denotes the elements ofPG with coefficient sum one which are of order a power ofp, it is shown that the elements of any subgroupH ofV _p (G) are linearly independent overP, and if in additionH is of orderp ⁿ , thenPGPH. As a consequence, the lattice of normal subgroups ofG and the abelianization of the normal subgroups ofG are determined byPG.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

Ideals in group rings of free products

LetG be a group andRG be its group ring. IfA is a nonzero ideal ofRG, we prove that for certain normal subgroupsH ofG, including all nontrivial subgroups ofG whenG is a free product,A∩RH≠0.     

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC _G(a) for 1≠a ∈A.     

A poisson formula for solvable Lie groups

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ^∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.     

On s-quasinormal and c-normal subgroups of a finite group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.     

Pebbling numbers of some graphs

Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that any distribution ofn pebbles onG allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphsG andH, f(G xH) ≤ f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham’s conjecture holds whenG andH are fan graphs or wheel graphs.     

Reconstructing graphs as subsumed graphs of hypergraphs,and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted . Ak-hypergraph G consists of avertex-set V(G) and anedgeset , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphG _v . The graphs {G _v νvεV(G)} aresubsumed byG. Each subsumed graphG _v is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG _v ≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG ^*, all of whose subsumed graphs are isomorphic toH, whereG andG ^* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.     

On the homology of certain spaces looped beyond their connectivity

Very little is known aboutH _*(Ωⁿ X) whenn is larger than the connectivity ofX. In this paper we calculate this whenX=Ω^∞ S ^∞ andn=1 or 2, and whenX=JU(q) or JSO(3) andn is arbitrary. Some information is also given whenX is a sphere. The authors were partially supported by the NSF.     

Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD _w .D ₁ is the only invariant control set, and the subsetW(S)={w:D _w =D ₁} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold. Research partially supported by CNPq grant n^o 50.13.73/91-8     

On the number of copies of one hypergraph in another

Given two hypergraphsH andG, letN(G, H) denote the number of subhypergraphs ofG isomorphic toH. LetN(l, H) denote the maximum ofN(G, H), taken over allG with exactlyl edges. In [1] Noga Alon analyzes the asymptotic behaviour ofN(l, H) forH a graph. We generalize this to hypergraphs: Theorem:For a hypergraph H with fractional cover number ρ*,N(G,H).=θ(lρ*) The interesting part of this is the upper bound, which is shown to be a simple consequence of an entropy lemma of J. Shearer. In a special case which includes graphs, we also provide a different proof using a hypercontractive estimate. Research supported in part by NSF.     

On the number of certain subgraphs contained in graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphsG, H, letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also, forl≧0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We determineN(l, H) precisely for alll≧0 whenH is a disjoint union of two stars, and also whenH is a disjoint union ofr≧3 stars, each of sizes ors+1, wheres≧r. We also determineN(l, H) for sufficiently largel whenH is a disjoint union ofr stars, of sizess ₁≧s ₂≧…≧s _r>r, provided (s ₁−s _r)²<s ₁+s _r−2r. We further show that ifH is a graph withk edges, then the ratioN(l, H)/l ^k tends to a finite limit asl→∞. This limit is non-zero iffH is a disjoint union of stars.     

Semiprime Goldie centralizers

LetG be a finite group of automorphisms acting on a ringR, andR ^G={fixed points ofG}. We show that under certain conditions onR andG, whenR ^Gis semiprime Goldie then so isR. In particular, ifa∈R is invertible anda ⁿ∈Z(R), thenR ^G,withG generated by the inner automorphism determined bya, is the centralizer ofa—C _R(a). The above result withR ^Greplaced byC _R(a) is shown without the assumption thata is invertible.     

Remarks on the wonderful compactification of semisimple algebraic groups

We prove that ifG is a semisimple algebraic group of adjoint type over the field of complex numbers,H is the subgroup of all fixed points of an involution σ ofG that is induced by an involution σ of the simply connected coveringĜ ofG, then the wonderful compactification of the homogeneous spaceG/H is isomorphic to the G.I.T quotientG ^ss (L)//H of the wonderful compactificationG ofG for a suitable choice of a line bundleL onG. We also prove a functorial property of the wonderful compactifications of semisimple algebraic groups of adjoint type.     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).