Butler groups of arbitrary cardinality

We show that in the constructible universe, the two usual definitions of Butler groups are equivalent for groups of arbitrarily large power. We also prove that Bext²(G, T) vanishes for every torsion-free groupG and torsion groupT. Furthermore, balanced subgroups of completely decomposable groups are Butler groups. These results have been known, under CH, only for groups of cardinalities ≤ ℵ_ω. Partial support by NSF is gratefully acknowledged. Partially supported by U.S.-Israel Binational Science Foundation.     

STACKED BASES FOR HOMOGENEOUS COMPLETELY DECOMPOSABLE GROUPS

Generalizing a theorem by P. Hill and C. Megibben, fixing a rational group R, we characterize by numerical invariants R-presentations of a group G, namely, short exact sequences of the form 0 → A → X → G → 0, where A and X are homogeneous completely decomposable groups of the same type R. This characterization sets afloat the class of the “uniquely R-presented groups”. This class is investigated in connection with the extension to arbitrary groups of the Warfield equivalence between categories of torsionfree abelian groups induced by the functors Hom(R, –) and R ? ?. As an application, the stacked bases theorem proved by J. Cohen and H. Gluck in 1970 is extended to arbitrary pairs of homogeneous completely decomposable abelian groups of the same type.

     

On groups of central type,non-degenerate and bijective cohomology classes

A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class [c] ∈ H ²(G, ℂ*) (G acts trivially on ℂ*). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation-theoretical properties. Suppose that a finite group Q acts on an abelian group A so that there exists a bijective 1-cocycle π ∈ Z ¹(Q,Ǎ), where Ǎ = Hom(A, ℂ*) is endowed with the diagonal Q-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in Z ²(G, ℂ*), where G:= A × Q. Hence, the semidirect product G is of central type. In this paper, we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class [π] ∈ H ¹(Q,Ǎ) as above, we construct non-degenerate classes [cπ] ∈ H ²(G,ℂ*) for certain extensions 1 → A → G → Q → 1 which are not necessarily split. We thus strictly extend the above family of central type groups.     

Identities of the soluble product of abelian groups

We consider the product G of abelian groups in the variety \mathfrakAⁿ \mathfrak{A}^n of soluble groups of length at most n. Provided that the abelian factors are decomposable into direct products of cyclic groups, we find necessary and sufficient conditions for G to generate the variety \mathfrakAⁿ \mathfrak{A}^n .     

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.     

Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties

Let H⊂G be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| _H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module , and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| _H of discrete series π for a symmetric space G/H is also given. Oblatum 2-X-1996 & 10-III-1997     

Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P（A） be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P（A）. In this paper, we characterize 3-regular compact graphs and prove that if G is a connected regular compact graph, G - v is also compact, and give a family of almost regular compact connected graphs.     

Bers-orlicz spaces on the product riemann surface

In this paper, the Bers-Orlicz spaces on the automorphic formA _α ^ϕ (G) (orEA _α ^ϕ (G)) andL _α ^ϕ (G) on the product Riemann surfaces are studied. We prove that eachf∈A _α ^ϕ (G) is a cusp form. Forf∈A _α ^ϕ (G), we give the reproducing formula. And, we give the projective operatorP _gga fromL _α ^ϕ (G) toA _α ^ϕ (G) toEA _α ^ϕ (G)). After giving some fundamental properties of the Poincaré series, we prove a dual theoremA _α ^ϕ (G)=(EA _α ^ϕ (G)). Supported by the National Nature Science Foundation of China     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

Derived length and products of conjugacy classes

Let G be a supersolvable group and A be a conjugacy class of G. Observe that for some integer η(AA ⁻¹) > 0, AA ⁻¹ = {ab ⁻¹: a, b ∈ A} is the union of η(AA ⁻¹) distinct conjugacy classes of G. Set C _G (A) = {g ∈ G: a ^g = a for all a ∈ A. Then the derived length of G/C _G (A) is less or equal than 2η(AA ⁻¹) − 1.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

On classification of the Butler and almost completely decomposable groups of finite rank

An approach to the study of torsion-free Abelian groups of finite rank developed by the author in an earlier paper is applied to smaller classes of Butler groups and of almost completely decomposable groups. Bibliography:3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 125–139.     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002     

Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

Let X be a block-rigid almost completely decomposable group of ring type with regulator A and p-primary regulator quotient X/A such that p ^l = exp X/A with natural l > 1. From the well-known fact p ^l End A ⊂ End X ⊂ End A it follows that End X = End X ∪ End A and p ^l End A = End X ∪ p ^l End A. Generalizing these, we determine the chain End X = ɛ _A ^(l) ⊂ ɛ _A ^(l−1) ⊂ ɛ _A ^(l−2) ⊂ ⋯ ⊂ ɛ _A ⁽¹⁾ ⊂ ɛ _A ⁽⁰⁾ = End A, satisfying p ^l−k ɛ _A ^(k) = End X ∪p ^l−k End A, and construct groups X _k ^′ and such that ɛ _A ^(k) = Hom , where k = 1, 2,..., l − 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 17–38, 2006.     

Varieties of almost polynomial growth: classifying their subvarieties

Let G be the infinite dimensional Grassmann algebra over a field F of characteristic zero and UT ₂ the algebra of 2 × 2 upper triangular matrices over F. The relevance of these algebras in PI-theory relies on the fact that they generate the only two varieties of almost polynomial growth, i.e., they grow exponentially but any proper subvariety grows polynomially. In this paper we completely classify, up to PI-equivalence, the associative algebras A such that A ∈ Var(G) or A ∈ Var(UT ₂).     

Local linear operators and multicontour solutions of homogeneous coupled map lattices

Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg _iB_ig_i ⁻¹ andA+B _i, whereg _i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB _i possess the following property: ‖B _iA⁻¹gB_jA⁻¹‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.