Nilpotent-by-Finite Cofinitary Groups

Throughout this paper, D denotes a division ring (possibly commutative)and V a left vector space over D, usually, but not exclusively,infinite-dimensional. We consider irreducible subgroups G ofGL(V) and are particularly interested in such G that containan element g the fixed-point set C_V(g) of which is non-zerobut finite-dimensional (over D). We then use this to deriveconclusions about cofinitary groups, an element g of GL(V) beingcofinitary if dim_DC_V(g) is finite, and a subgroup of GL(V) beingcofinitary if all its non-identity elements are cofinitary. Suppose that G is a cofinitary subgroup of GL(V). There aretwo extreme cases. If dim_DV is finite the cofinitary conditionis vacuous. At the other extreme, if G acts fixed-point freelyon V then the fixed-point sets C_V(g) for gG\1 are as small aspossible, namely {0}. Work of Blichfeldt and his successorsshows that certain irreducible linear groups G of dimensionat least 2 over, for example, the complexes are always imprimitive.This is the case if G is nilpotent, or supersoluble, or metabelian.Apart from the two extreme cases, the same is frequently truefor irreducible cofinitary subgroups G of GL(V). For example,this is the case if G is finitely generated nilpotent [9, 1.2]or more generally if G is supersoluble [10, 1.1], but not ingeneral if G is metabelian [10, 7.1] or parasoluble (a groupG is parasoluble if it has a normal series of finite lengthsuch that every subgroup of each of its factors is Abelian andnormalised by G) (see [10, 7.2]). Further, it is also the caseif G is Abelian-by-finite [10, 3.4], and every supersolublegroup is finitely generated and nilpotent-by-finite. Collectively,these results suggest that one should consider nilpotent-by-finitegroups.     

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_q[G] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_q[G] is Noetherian recovers aresult due to Joseph [10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.     

The Transformation Groups Whose C*-Algebras are Fell Algebras

Let (G, X) be a locally compact transformation group in whichG acts freely on X. We show that the associated transformation-groupC*-algebra C₀(X) G is a Fell algebra if and only if X is aCartan G-space.     

Noether Numbers for Subrepresentations of Cyclic Groups of Prime Order

Let W be a finite-dimensional Z/p-module over a field, k, ofcharacteristic p. The maximum degree of an indecomposable elementof the algebra of invariants, k[W]^Z/p, is called the Noethernumber of the representation, and is denoted by rß(W).A lower bound for rß(W) is derived, and it is shownthat if U is a Z/p submodule of W, then rß(U) rß(W).Aset of generators, in fact a SAGBI basis, is constructed fork[V₂ V₃]^Z/p, where V_n is the indecomposable Z/p-module of dimensionn. 2000 Mathematics Subject Classification 13A50, 20J06.     

The Non-Finite Presentability of the Automorphism Group of the Free Z-Group of Rank Two

Let G be a group, and let F_n[G] be the free G-group of rankn. Then F_n[G] is just the natural non-abelian analogue of thefree ZG-module of rank n, and correspondingly the group _n(G)of equivariant automorphisms of F_n[G] is a natural analogueof the general linear group GL_n(ZG). The groups _n(G) have beenstudied recently in [4, 8, 5]. In particular, in [5] it wasshown that if G is not finitely presentable (f.p.) then neitheris _n(G), and conversely, that _n(G) is f.p. if G is f.p. andn2. It is a common phenomenon that for small ranks the automorphismgroups of free objects may behave unstably (see the survey article[2]), and the main aim of the present paper is to show thatthis turns out to be the case for the groups ₂(G).     

On locally finite groups with a small centralizer of a four-subgroup

The main result of the paper is the following theorem. Let G be a locally finite group having a four-subgroup V such that C _G (V) is finite. Suppose that V contains two involutions v ₁ and v ₂ such that the centralizers C _G (v ₁) and C _G (v ₂) have finite exponent. Then G is almost locally soluble and [G, V]′ has finite exponent. Since [G, V] has finite index in G, the result gives a fairly detailed information about the structure of G.     

A Bound on the Presentation Rank of a Finite Group

Let G be a finite group, and let I_G be the augmentation idealof ZG. We denote by d(G) the minimum number of generators forthe group G, and by d(I_G) the minimum number of elements ofI_G needed to generate I_G as a G-module. The connection betweend(G) and d(I_G) is given by the following result due to Roggenkamp]14]: where pr(G) is a non-negative integer, called the presentationrank of G, whose definition comes from the study of relationmodules (see [4] for more details). 1991 Mathematics SubjectClassification 20D20.     

Locally Finite Finitary Skew Linear Groups

Let V be a vector space over the division ring D of infinitedimension. We study locally finite, primitive groups G of finitarylinear automorphisms of V. We show that the derived group G'of G is infinite, simple, and lies in every non-trivial normalsubgroup of G, and that G' G Aut G'. Moreover if char D =0, then G is either the finitary symmetric group or the alternatinggroup on some infinite set. If D is commutative, that is, ifD is a field, then all these results are known and are the consequenceof the collective work of a number of people: in particular(in alphabetical order) V. V. Belyaev, J. I. Hall, F. Leinen,U. Meierfrankenfeld, R. E. Phillips, O. Puglisi, A. Radfordand quite probably others. 2000 Mathematics Subject Classification:20H25, 20H20, 20F50.     

On Self-Contragredient Genera Of Z[G]-Lattices

If G is a finite group and V is a finite-dimensional Q[G]-module,V is isomorphic to its contragredient module V*. In general,V need not contain any Z[G]-lattice which is locally isomorphicto its contragredient lattice. Nevertheless, it turns out thatfor every V there exists another Q[G]-module V' such that bothV' and V V' contain Z[G]-lattices which are locally isomorphicto their contragredient lattices. 2000 Mathematics Subject Classification20C10, 20C05.     

Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

A Characterization of Finite Soluble Groups by Laws in Two Variables

Define a sequence (s_n) of two-variable words in variables x,y as follows: s₀(x, y) = x, s_n+1(x,y)=[s_n(x, y]^–y, s_n(x,y)for n 0. It is shown that a finite group G is soluble if andonly if s_n is a law of G for all but finitely many values ofn. 2000 Mathematics Subject Classification 20D10, 20D06.     

Isomorphisms of Group Algebras

Let G₁ and G₂ be locally compact groups. If T is an algebraisomorphism of L¹(G₁) onto L¹(G₂) with ||T|| (1+3), then G₁and G₂ are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.     

Full Crossed Products by Coactions, C0(X)-Algebras and C*-Bundles

Let be a coaction of a locally compact group G on a C*-algebraA. We show that if I is a -invariant ideal in A, then for full crossed products, as Landstadet al. have done for spatial crossed products by coactions.We prove that for suitable coactions, the crossed products ofC₀(X)-algebras are again C₀(X)-algebras, and the crossed productsof continuous C*-bundles by a locally compact group are againcontinuous C*-bundles. 1991 Mathematics Subject Classification46L55.     

Linear Independence of Hecke Operators in the Homology of X0(N)

In Merel's recent proof [7] of the uniform boundedness conjecturefor the torsion of elliptic curves over number fields, a keystep is to show that for sufficiently large primes N, the Heckeoperators T₁, T₂, ..., T_D are linearly independent in theiractions on the cycle e from 0 to i in H₁(X₀(N) (C), Q). In particular,he shows independence when max(D⁸, 400D⁴) < N/(log N)⁴. Inthis paper we use analytic techniques to show that one can chooseD considerably larger than this, provided that N is large.     

One Cubic Diophantine Inequality

Suppose that G(x) is a form, or homogeneous polynomial, of odddegree d in s variables, with real coefficients. Schmidt [15]has shown that there exists a positive integer s₀(d), whichdepends only on the degree d, so that if s s₀(d), then thereis an x Z^s\{0} satisfying the inequality |G(x)|<1. (1) In other words, if there are enough variables, in terms of thedegree only, then there is a nontrivial solution to (1). Lets₀(d) be the minimum integer with the above property. In thecourse of proving this important result, Schmidt did not explicitlygive upper bounds for s₀(d). His methods do indicate how todo so, although not very efficiently. However, in fact muchearlier, Pitman [13] provided explicit bounds in the case whenG is a cubic. We consider a general cubic form F(x) with realcoefficients, in s variables, and look at the inequality |F(x)|<1. (2) Specifically, Pitman showed that if s(1314)²⁵⁶–1, (3) then inequality (2) is non-trivially soluble in integers. Wepresent the following improvement of this bound.     

Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.     

Nest Representations of Directed Graph Algebras

This paper is a comprehensive study of the nest representationsfor the free semigroupoid algebra L_G of a countable directedgraph G as well as its norm-closed counterpart, the tensor algebraT⁺(G). We prove that the finite-dimensional nest representations separatethe points in L_G, and a fortiori, in T⁺(G). The irreduciblefinite-dimensional representations separate the points in L_Gif and only if G is transitive in components (which is equivalentto being semisimple). Also the upper triangular nest representationsseparate points if and only if for every vertex x T(G) supportinga cycle, x also supports at least one loop edge. We also study faithful nest representations. We prove that L_G(or T⁺(G) admits a faithful irreducible representation if andonly if G is strongly transitive as a directed graph. More generally,we obtain a condition on G which is equivalent to the existenceof a faithful nest representation. We also give a conditionthat determines the existence of a faithful nest representationfor a maximal type N nest. 2000 Mathematics Subject Classification47L80, 47L55, 47L40.     

Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type

Let G = G(q) be a finite almost simple exceptional group ofLie type over the field of q elements, where q = p^a and p isprime. The main result of the paper determines all maximal subgroupsM of G(q) such that M is an almost simple group which is alsoof Lie type in characteristic p, under the condition that rank(M)> rank(G). The conclusion is that either M is a subgroupof maximal rank, or it is of the same type as G over a subfieldof F_q, or (G, M) is one of (, F₄(q)), (, C₄(q)), (E₇(q),³D₄(q)). This completes work of the first author with Saxl andTesterman, in which the same conclusion was obtained under someextra assumptions.     

The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and Va finite-dimensional FG-module. Let L(V) denote the free Liealgebra on V regarded as an FG-submodule of the free associativealgebra (or tensor algebra) T(V). For each positive integerr, let L^r (V) and T^r (V) be the rth homogeneous components ofL(V) and T(V), respectively. Here L^r (V) is called the rth Liepower of V. Our main result is that there are submodules B₁,B₂, ... of L(V) such that, for all r, B_r is a direct summandof T^r(V) and, whenever m 0 and k is not divisible by p, themodule is the direct sum of , . Thus every Lie power is a direct sum of Lie powers of p-powerdegree. The approach builds on an analysis of T^r (V) as a bimodulefor G and the Solomon descent algebra. 2000 Mathematics SubjectClassification 17B01 (primary), 20C07, 20C20 (secondary).