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.     

Abstract Homomorphisms of Algebraic Groups

In this paper we are concerned with abstract homomorphisms fromone algebraic group to another. That is, group homomorphismswhere there is no assumption of rationality. Let K be an infiniteperfect field and L an algebraically closed field of the samecharacteristic. Throughout G(K) will denote a Chevalley groupof universal type over K.     

ZERO-SUM PROBLEMS IN FINITE ABELIAN GROUPS AND AFFINE CAPS

For a finite abelian group G, let (G) denote the smallest integer l such that every sequence Sover G of length | S| l has a zero-sum subsequence of lengthexp (G). We derive new upper and lower bounds for (G), and all our bounds are sharp for special typesof groups. The results are not restricted to groups G of theform , but they respect the structure of the group. In particular, we show for all odd n, which is sharp if n is a power of3. Moreover, we investigate the relationship between extremalsequences and maximal caps in finite geometry.     

Extensions of Asymptotic Fields Via Meromorphic Functions

An asymptotic field is a special type of Hardy field in which,modulo an oracle for constants, one can determine asymptoticbehaviour of elements. In a previous paper, it was shown inparticular that limits of real Liouvillian functions can therebybe computed. Let denote an asymptotic field and let f . Weprove here that if G is meromorphic at the limit of f (whichmay be infinite) and satisfies an algebraic differential equationover R(x), then (G o f) is an asymptotic field. Hence it ispossible (modulo an oracle for constants) to compute asymptoticforms for elements of (G o f). An example is given to show thatthe result may fail if G has an essential singularity at limf.     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.     

On the Fitting Height of a Soluble Group that is Generated by a Conjugacy Class

Let G be a finite group and suppose that P is a soluble {2,3}'-subgroup of G. The reader will lose only a little by assumingthat P is a subgroup of prime order p > 3. _G(P)={AG|A is soluble and A=P,P^a for some a A This set is partially ordered by inclusion and we let denote the set of maximal members of _G(P).     

The Series of Norms in a Soluble p-Group

The norm of a group G is the subgroup of elements of G whichnormalise every subgroup of G. We shall denote it (G). An ascendingseries of subgroups _i(G) in G may be defined recursively by:₀(G) = 1 and, for i 0, _i+1(G)/_i(G) = (G/_i(G)). For each i,the section _i+1(G)/i(G) clearly contains the centre of the groupG/_i(G). A result of Schenkman [8] gives a very close connectionbetween this norm series and the upper central series: _i(G) _i(G) _2i(G). 1991 Mathematics Subject Classification 20E15.     

Rank of Elliptic Curves over almost Separably Closed Fields

Let E be an elliptic curve over a finitely generated infinitefield K. Let K^s denote a separable closure of K, an elementof the Galois group G_K=Gal(K^s/K), and K^s() the invariant subfieldof K^s. If the characteristic of K is not 2 and belongs to asuitable open subgroup of G_K, then E(K^s()) has infinite rank.2000 Mathematics Subject Classification 11G05.     

Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in H^q(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.     

Crossed Product Orders over Valuation Rings

Let V be a commutative valuation domain of arbitrary Krull-dimension(rank), with quotient field F, and let K be a finite Galoisextension of F with group G, and S the integral closure of Vin K. If, in the crossed product algebra K * G, the 2-cocycletakes values in the group of units of S, then one can form,in a natural way, a ‘crossed product order’ S *G K * G. In the light of recent results by H. Marubayashi andZ. Yi on the homological dimension of crossed products, thispaper discusses necessary and/or sufficient valuation-theoreticconditions, on the extension K/F, for the V-order S * G to besemihereditary, maximal or Azumaya over V. 2000 MathematicsSubject Classification 16H05, 16S35.     

The spine of a Fourier-Stieltjes algebra

We define the spine A *(G) of the Fourier–Stieltjes algebraB (G) of a locally compact group G. This algebra encodes informationabout much of the fine structure of B (G), particularly informationabout certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, thatof non-quotient locally precompact topologies on G. We computethe spine's spectrum G*, which admits a semi-group structure.We discuss homomorphisms from A *(G) to B (H) where H is anotherlocally compact group; and we show that A *(H) contains theimage of every completely bounded homomorphism from the Fourieralgebra A (H) of any amenable group G. We also show that A *(G)contains all of the idempotents in B (G). Finally, we computeexamples for vector groups, abelian lattices, minimally almostperiodic groups and the (ax + b)-group; and we explore the complexityof A *(G) for the discrete rational numbers and free groups.     

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].     

Free Lie Algebras and Adams Operations

Let G be a group and K a field. For any finite-dimensional KG-moduleV and any positive integer n, let L^n(V) denote the nth homogeneouscomponent of the free Lie K-algebra generated by (a basis of)V. Then L^n(V) can be considered as a KG-module, called the nthLie power of V. The paper is concerned with identifying thismodule up to isomorphism. A simple formula is obtained whichexpresses L^n(V) in terms of certain linear functions on theGreen ring. When n is not divisible by the characteristic ofK these linear functions are Adams operations. Some resultsare also obtained which clarify the relationship between Adamsoperations defined by means of exterior powers and symmetricpowers and operations introduced by Benson. Some of these resultsare put into a more general setting in an appendix by StephenDonkin.     

On Waring's Problem in Number Fields

Let K be an algebraic number field of degree n over the rationals,and denote by J_k the subring of K generated by the kth powersof the integers of K. Then G_K(k) is defined to be the smallests1 such that, for all totally positive integers vJ_k of sufficientlylarge norm, the Diophantine equation (1.1) is soluble in totally non-negative integers _i of K satisfying N(_i)<<N(v)^1/k (1is). (1.2) In (1.2) and throughout this paper, all implicit constants areassumed to depend only on K, k, and s. The notation G_K(k) generalizesthe familiar symbol G(k) used in Waring's problem, since wehave G_Q(k) = G(k). By extending the Hardy–Littlewood circle method to numberfields, Siegel [8, 9] initiated a line of research (see [1–4,11]) which generalized existing methods for treating G(k). Thistypically led to upper bounds for G_K(k) of approximate strengthnB(k), where B(k) was the best contemporary upper bound forG(k). For example, Eda [2] gave an extension of Vinogradov'sproof (see [13] or [15]) that G(k)(2+o(1))k log k. The presentpaper will eliminate the need for lengthy generalizations assuch, by introducing a new and considerably shorter approachto the problem. Our main result is the following theorem.     

Spectral Synthesis and Operator Synthesis for Compact Groups

Let G be a compact group and C(G) be the C^*-algebra of continuouscomplex-valued functions on G. The paper constructs an imbeddingof the Fourier algebra A(G) of G into the algebra V(G) = C(G)^hC(G)(Haagerup tensor product) and deduces results about parallelspectral synthesis, generalizing a result of Varopoulos. Itthen characterizes which diagonal sets in G x G are sets ofoperator synthesis with respect to the Haar measure, using thedefinition of operator synthesis due to Arveson. This resultis applied to obtain an analogue of a result of Froelich: atensor formula for the algebras associated with the pre-ordersdefined by closed unital subsemigroups of G.     

Sub-Laplacians of Holomorphic Lp-Type on Exponential Solvable Groups

Let L denote a right-invariant sub-Laplacian on an exponential,hence solvable Lie group G, endowed with a left-invariant Haarmeasure. Depending on the structure of G, and possibly alsothat of L, L may admit differentiable L^p-functional calculi,or may be of holomorphic L^p-type for a given p 2. ‘HolomorphicL^p-type’ means that every L^p-spectral multiplier for Lis necessarily holomorphic in a complex neighbourhood of somenon-isolated point of the L²-spectrum of L. This can in factonly arise if the group algebra L¹(G) is non-symmetric. Assume that p 2. For a point in the dual g^* of the Lie algebrag of G, denote by ()=Ad^*(G) the corresponding coadjoint orbit.It is proved that every sub-Laplacian on G is of holomorphicL^p-type, provided that there exists a point g^* satisfying Boidol'scondition (which is equivalent to the non-symmetry of L¹(G)),such that the restriction of () to the nilradical of g is closed.This work improves on results in previous work by Christ andMüller and Ludwig and Müller in twofold ways: on theone hand, no restriction is imposed on the structure of theexponential group G, and on the other hand, for the case p>1,the conditions need to hold for a single coadjoint orbit only,and not for an open set of orbits. It seems likely that the condition that the restriction of ()to the nilradical of g is closed could be replaced by the weakercondition that the orbit () itself is closed. This would thenprove one implication of a conjecture by Ludwig and Müller,according to which there exists a sub-Laplacian of holomorphicL¹ (or, more generally, L^p) type on G if and only if there existsa point g^* whose orbit is closed and which satisfies Boidol'scondition.     

Homological Properties of Modules Over Group Algebras

Let G be a locally compact group, and let L¹ (G) be the Banachalgebra which is the group algebra of G. We consider a varietyof Banach left L¹ (G)-modules over L¹ (G), and seek to determineconditions on G that determine when these modules are eitherprojective or injective or flat in the category. The answerstypically involve G being compact or discrete or amenable. Forexample, in the case where G is discrete and 1 < p < ,we find that the module ^p (G) is injective whenever G is amenable,and that, if it is amenable, then G is ‘pseudo-amenable’,a property very close to that of amenability. 2000 MathematicsSubject Classification 46H25, 43A20.     

The Natural Morphisms between Toeplitz Algebras on Discrete Groups

Let G be a discrete group and (G, G₊) be a quasi-ordered group.Set G₊(G₊)^–1 and G₁= (G₊\){e}. Let F^G₁(G) andF^G₊(G) be the corresponding Toeplitz algebras. In the paper,a necessary and sufficient condition for a representation ofF^G₊(G) to be faithful is given. It is proved that when G isabelian, there exists a natural C*-algebra morphism from F^G₁(G)to F^G₊(G). As an application, it is shown that when G = Z² andG₊ = Z₊ x Z, the K-groups K₀(F^G₁(G)) Z², K₁(F^G₁(G)) Z andall Fredholm operators in F^G₁(G) are of index zero.     

Operator amenability of Fourier-Stieltjes algebras, II

We give an example of a non-compact, locally compact group Gsuch that its Fourier–Stieltjes algebra B (G) is operatoramenable. Furthermore, we characterize those G for which A *(G),the spine of B (G) as introduced by M. Ilie and N. Spronk, isoperator amenable and show that A *(G) is operator weakly amenablefor each G.     

Barely Transitive and Heineken Mohamed Groups

A negative answer to the Kuro–ernikov Question 21 in [7],whether a group satisfying the normalizer condition is hypercentral,was given by Heineken and Mohamed in 1968 [6]. They constructedgroups G satisfying: (i) G is a locally finite p-group for a prime p, (ii) G/G'C_p and G' is countable elementary abelian, (iii) every proper subgroup of G is subnormal and nilpotent, (iv) Z(G)={1}, (v) the set of normal subgroups of G contained in G' is linearlyordered by set inclusion, see [3, p. 334], (vi) KG' is a proper subgroup in G for every proper subgroupK of G, see [6, Lemma 1(a)].