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1.
Let G be a finite group, and let IG be the augmentation idealof ZG. We denote by d(G) the minimum number of generators forthe group G, and by d(IG) the minimum number of elements ofIG needed to generate IG as a G-module. The connection betweend(G) and d(IG) 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. 相似文献
2.
When G is abelian and l is a prime we show how elements of therelative K-group K0(Zl[G], Ql 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. 相似文献
3.
Cutolo G.; Khukhro E. I.; Lennox J. C.; Rinauro S.; Smith H.; Wiegold James 《Bulletin London Mathematical Society》1997,29(5):563-570
For n a positive integer, a group G is called core-n if H/HGhas order at most n for every subgroup H of G (where HG 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. 相似文献
4.
Let G be an almost simple algebraic group defined over Fp 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 correspondingKazhdanLusztig 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. 相似文献
5.
For a semigroup S, the finitary power semigroup of S, denotedPf(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that Pf(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary). 相似文献
6.
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. 相似文献
7.
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 Qp, 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. 相似文献
8.
A Uniqueness Problem in Valued function Fields of Conics 总被引:1,自引:0,他引:1
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]. 相似文献
9.
Given any sequence of non-abelian finite simple primitive permutationgroups Sn, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...Sn Sn1 ... S0. It follows that the upper compositionfactors of G are exactly the groups Sn. By suitably choosingthe sequence Sn 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. 相似文献
10.
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/CV(G) need not be completelyreducible, even if dimDV is finite. Moreover, if F is a field,then V and V/CV(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. 相似文献
11.
Let P be an n-dimensional polytope admitting a finite reflectiongroup G as its symmetry group. Consider the set HP(k) of allcontinuous functions on Rn satisfying the mean value propertywith respect to the k-skeleton P(k) of P, as well as the setHG of all G-harmonic functions. Then a necessary and sufficientcondition for the equality HP(k) = HG is given in terms of adistinguished invariant basis, called the canonical invariantbasis, of G. 1991 Mathematics Subject Classification 20F55,52B15. 相似文献
12.
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(CG(g)) of subspaceswhich are fixed points of elements of CG(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-atomproperty of L(CG(g)), which asserts that if [X1,..., Xk} isany set of elements of L(CG(g)) which are maximal (set theoretically)then their setwise intersection lies in L(CG(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40. 相似文献
13.
Let G be a discrete group and (G, G+) be a quasi-ordered group.Set G+(G+)1 and G1= (G+\){e}. Let FG1(G) andFG+(G) be the corresponding Toeplitz algebras. In the paper,a necessary and sufficient condition for a representation ofFG+(G) to be faithful is given. It is proved that when G isabelian, there exists a natural C*-algebra morphism from FG1(G)to FG+(G). As an application, it is shown that when G = Z2 andG+ = Z+ x Z, the K-groups K0(FG1(G)) Z2, K1(FG1(G)) Z andall Fredholm operators in FG1(G) are of index zero. 相似文献
14.
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. 相似文献
15.
Let G be a group, and let Fn[G] be the free G-group of rankn. Then Fn[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 Fn[G] is a natural analogueof the general linear group GLn(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). 相似文献
16.
Gluing torsion endo-permutation modules 总被引:1,自引:0,他引:1
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 MQ in the Dade groupD(NP(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 MQ is only given for centric subgroups Q of P. Moreover,general results about biset functors and the Dade functor aregiven in two appendices. 相似文献
17.
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 (ZbxZb)Aut0 E0 where b is theself-intersection number of E0, ZbxZb 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 naturalsection 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. 相似文献
18.
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 m2,p(G) = 3. 2000 MathematicsSubject Classification 20D05. 相似文献
19.
It is proved that if PG(s) has an Euler product expansion withall factors of the form where each qi is a prime power, then G is soluble. 2000 MathematicsSubject Classification 20P05, 20F05, 11M41. 相似文献
20.
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, ..., Hn are finitely generatedsubgroups of G, then the product subset H1 H2 ... Hn 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]). 相似文献