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.     

Relative K0, Annihilators, Fitting Ideals and Stickelberger Phenomena

When G is abelian and l is a prime we show how elements of therelative K-group K₀(Z_l[G], Q_l give rise to annihilator/Fittingideal relations of certain associated Z[G]-modules. Examplesof this phenomenon are ubiquitous. Particularly, we give examplesin which G is the Galois group of an extension of global fieldsand the resulting annihilator/Fitting ideal relation is closelyconnected to Stickelberger's Theorem and to the conjecturesof Coates and Sinnott, and Brumer. Higher Stickelberger idealsare defined in terms of special values of L-functions; whenthese vanish we show how to define fractional ideals, generalisingthe Stickelberger ideals, with similar annihilator properties.The fractional ideal is constructed from the Borel regulatorand the leading term in the Taylor series for the L-function.En route, our methods yield new proofs, in the case of abeliannumber fields, of formulae predicted by Lichtenbaum for theorders of K-groups and étale cohomology groups of ringsof algebraic integers. 2000 Mathematics Subject Classification11G55, 11R34, 11R42, 19F27.     

Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H_Ghas order at most n for every subgroup H of G (where H_G is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.     

Filtrations on G1T-Modules

Let G be an almost simple algebraic group defined over F_p forsome prime p. Denote by G₁ the first Frobenius kernel in G andlet T be a maximal torus. In this paper we study certain Jantzentype filtrations on various modules in the representation theoryof G₁T. We have such filtrations on the baby Verma modules Z,where is a character of T. They are obtained via a certaindeformation of the natural homomorphism from Z into its contravariantdual Z. Using the same deformation we construct for each projectiveG₁T-module Q a filtration of the vector space . We then prove that this filtration may also bedescribed in terms of the above-mentioned homomorphism Z() Z() and this leads us to a sum formula for our filtrations.When Q is indecomposable with highest weight in the bottom alcove(with respect to some special point) we are able to computethe filtrations on F(Q) explicitly for all . This is then thestarting point of an induction which proceeds via wall crossingsto higher alcoves. If our filtrations behave as expected undersuch wall crossings then we obtain a precise relation betweenthedimensions of the layers in the filtrations of F(Q) for an arbitraryindecomposable projective Q and the coefficients in the correspondingKazhdan–Lusztig polynomials. We conclude the paper byproving that the above results in the G₁T theory have some analoguesin the representation theory of G (where, however, we have towork with representations of bounded highest weights) and thecorresponding theory for quantum groups at roots of unity. Theseresults extend previous work by the first author. 2000 MathematicsSubject Classification: 20G05, 20G10, 17B37.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

Morita Equivalent Principal 3-Blocks of the Chevalley Groups G2(q)

The principal 3-block of a Chevalley group G₂(q) with q a powerof 2 satisfying q 2 or 5 mod 9 and the principal 3-block ofG₂(2) are Morita equivalent. 2000 Mathematical Subject Classification:20C05, 20C20, 20C33.     

Mod p Reducibility of Unramified Representations of Finite Groups of Lie Type

Dedicated to the memory of Professor A. I. Kostrikin The main problem under discussion is to determine, for quasi-simplegroups of Lie type G, irreducible representations of G thatremain irreducible under reduction modulo the natural primep. The method is new. It works only for p >3 and for representations that can be realized over an unramified extension of Q_p, thefield of p -adic numbers. Under these assumptions, the mainresult says that the trivial and the Steinberg representationsof G are the only representations in question provided G isnot of type A1. This is not true for G=SL(2, p). The paper containsa result of independent interest on infinitesimally irrreduciblerepresentations of G over an algebraically closed field ofcharacteristic p. Assuming that G is not of rank 1 and G G₂(5),it is proved that either the Jordan normal form of a root elementcontains a block of size d with 1<d<p -1 or the highestweight of is equal to p -1 times the sum of the fundamentalweights. 2000 Mathematical Subject Classification: 20C33, 20G15.     

A Uniqueness Problem in Valued function Fields of Conics

Let v₀ be a valuation of a field K₀ with value group G₀. LetK be a function field of a conic over K₀, and let v be an extensionof v₀ to K with value group G such that G/G₀ is not a torsiongroup. Suppose that either (K₀, v₀) is henselian or v₀ is ofrank 1, the algebraic closure of K₀ in K is a purely inseparableextension of K₀, and G₀ is a cofinal subset of G. In this paper,it is proved that there exists an explicitly constructible elementt in K, with v(t) non-torsion modulo G₀ such that the valuationof K₀(t), obtained by restricting v, has a unique extensionto K. This generalizes the result proved by Khanduja in theparticular case, when K is a simple transcendental extensionof K₀ (compare [4]). The above result is an analogue of a resultof Polzin proved for residually transcendental extensions [8].     

The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.     

The Complete Reducibility of Locally Completely Reducible Finitary Linear Groups

Let V be a vector space over some division ring D, and G a finitarysubgroup of GL(V). If G is locally completely reducible, thenthe D-G modules V, [V, G] and V/C_V(G) need not be completelyreducible, even if dim_DV is finite. Moreover, if F is a field,then V and V/C_V(G) need not be completely reducible. We provehere that if D is a finite-dimensional division algebra andG is locally completely reducible, then [V, G] is always a completelyreducible D-G module. 1991 Mathematics Subject Classification20H25.     

Invariants of Finite Reflection Groups and the Mean Value Problem for Polytopes

Let P be an n-dimensional polytope admitting a finite reflectiongroup G as its symmetry group. Consider the set H_P(k) of allcontinuous functions on Rⁿ satisfying the mean value propertywith respect to the k-skeleton P(k) of P, as well as the setH_G of all G-harmonic functions. Then a necessary and sufficientcondition for the equality H_P(k) = H_G is given in terms of adistinguished invariant basis, called the canonical invariantbasis, of G. 1991 Mathematics Subject Classification 20F55,52B15.     

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.     

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.     

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.     

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).     

Gluing torsion endo-permutation modules

Let k be a field of characteristic p, and let P be a finitep-group, where p is an odd prime. In this paper, we considerthe problem of gluing compatible families of endo-permutationmodules: being given a torsion element M_Q in the Dade groupD(N_P(Q)/Q), for each non-trivial subgroup Q of P, subject toobvious compatibility conditions, we show that it is alwayspossible to find an element M in the Dade group of P such that for all Q, but that Mneed not be a torsion element of D(P). The obstruction to thisis controlled by an element in the zeroth cohomology group over₂ of the poset of elementary abelian subgroups of P of rankat least 2. We also give an example of a similar situation,when M_Q is only given for centric subgroups Q of P. Moreover,general results about biset functors and the Dade functor aregiven in two appendices.     

Symmetries of Surface Singularities

The study of reductive group actions on a normal surface singularityX is facilitated by the fact that the group Aut X of automorphismsof X has a maximal reductive algebraic subgroup G which containsevery reductive algebraic subgroup of Aut X up to conjugation.If X is not weighted homogeneous then this maximal group G isfinite (Scheja, Wiebe). It has been determined for cusp singularitiesby Wall. On the other hand, if X is weighted homogeneous butnot a cyclic quotient singularity then the connected componentG₁ of the unit coincides with the C* defining the weighted homogeneousstructure (Scheja, Wiebe, Wahl). Thus the main interest liesin the finite group G/G₁. Not much is known about G/G₁. Ganterhas given a bound on its order valid for Gorenstein singularitieswhich are not log-canonical. Aumann-Körber has determinedG/G₁ for all quotient singularities. We propose to study G/G₁ through the action of G on the minimalgood resolution of X. If X is weightedhomogeneous but not a cyclic quotient singularity, let E₀ bethe central curve of the exceptional divisor of . We show that the natural homomorphism GAut E₀ haskernel C* and finite image. In particular, this re-proves therest of Scheja, Wiebe and Wahl mentioned above. Moreover, itallows us to view G/G₁ as a subgroup of Aut E₀. For simple ellipticsingularities it equals (Z_bxZ_b)Aut₀ E₀ where –b is theself-intersection number of E₀, Z_bxZ_b is the group of b-torsionpoints of the elliptic curve E₀ acting by translations, andAut₀ E₀ is the group of automorphisms fixing the zero elementof E₀. If E₀ is rational then G/G₁ is the group of automorphismsof E₀ which permute the intersection points with the branchesof the exceptional divisor while preserving the Seifert invariantsof these branches. When there are exactly three branches weconclude that G/G₁ is isomorphic to the group of automorphismsof the weighted resolution graph. This applies to all non-cyclicquotient singularities as well as to triangle singularities.We also investigate whether the maximal reductive automorphismgroup is a direct product GG₁xG/G₁. This is the case, for instance,if the central curve E₀ is rational of even self-intersectionnumber or if X is Gorenstein such that its nowhere-zero 2-form has degree ±1. In the latter case there is a ‘natural’section G/G₁G of GG/G₁ given by the group of automorphisms inG which fix . For a simple elliptic singularity one has GG₁xG/G₁if and only if –E₀ · E₀ = 1.     

Toward a Characterization of Conway's Group Co3

In this paper, Conway's group, Co₃, is characterized among simplegroups of even type with e(G) = 3, by a restriction on the p-localstructure for some odd prime p for which m_2,p(G) = 3. 2000 MathematicsSubject Classification 20D05.     

Recognizing Soluble Groups from their Probabilistic Zeta Functions

It is proved that if P_G(s) has an Euler product expansion withall factors of the form where each q_i is a prime power, then G is soluble. 2000 MathematicsSubject Classification 20P05, 20F05, 11M41.     

The Product Separability of the Generalized Free Product of Cyclic Groups

Let G be a group endowed with its profinite topology, then Gis called product separable if the profinite topology of G isHausdorff and, whenever H₁, H₂, ..., H_n are finitely generatedsubgroups of G, then the product subset H₁ H₂ ... H_n is closedin G. In this paper, we prove that if G=FxZ is the direct productof a free group and an infinite cyclic group, then G is productseparable. As a consequence, we obtain the result that if Gis a generalized free product of two cyclic groups amalgamatinga common subgroup, then G is also product separable. These resultsgeneralize the theorems of M. Hall Jr. (who proved the conclusionin the case of n=1, [3]), and L. Ribes and P. Zalesskii (whoproved the conclusion in the case of that G is a finite extensionof a free group, [6]).