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.     

On the Efficiency of Coxeter Groups

If G is a finitely presented group and K is any (G,2)-complex(that is, a finite 2-complex with fundamental group G), thenit is well known that X(K) (G), where (G) = 1–rk H₁G+ dH₂G. We define (G) to be min{(K): K a (G, 2)-complex}, andwe say that G is efficient if (G)=(G). In this paper we givesufficient conditions for a Coxeter group to be efficient (Theorem4.2). We also give examples of inefficient Coxeter groups (Theorem5.1). In fact, we give an infinite family G_n(n = 2, 3, 4, ...)of Coxeter groups such that (G_n)–(G_n) as n . 1991 MathematicsSubject Classification 20F05, 20F55.     

A Liouville Theorem for Matrix-Valued Harmonic Functions on Nilpotent Groups

Let be a non-degenerate positive M_n-valued measure on a locallycompact group G with |||| = 1. An M_n-valued Borel function fon G is called -harmonic if for all x G. Given such a function f which is bounded and leftuniformly continuous on G, it is shown that every central elementin G is a period of f. Further, it is shown that f is constantif G is nilpotent or central. 2000 Mathematics Subject Classification31C05, 43A05, 45E10, 46G10.     

On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

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.     

On Translation Functors for General Linear and Symmetric Groups

In this paper we study the rational representation theory ofthe general linear group G = GL_n(F) over an algebraically closedfield F of characteristic p. Given Z/pZ, we define functorsTr and Tr, which, roughly speaking, are given by tensoring withthe natural G-module V and its dual V* respectively, and thenprojecting onto certain blocks determined by the residue . Infact, these functors can be viewed as special cases of Jantzen'stranslation functors. We prove a number of fundamental propertiesabout these functors and also certain closely related functorsthat arise in the modular representation theory of the symmetricgroup. 1991 Mathematics Subject Classification: 20G05, 20C05.     

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.     

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.     

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.     

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.     

Decompositions of Complete Graphs

If s₁, s₂, ..., s_t are integers such that n – 1 = s₁ +s₂ + ... + s_t and such that for each i (1 i t), 2 s_i n –1 and s_in is even, then K_n can be expressed as the union G₁G₂...G_tof t edge-disjoint factors, where for each i, G_i is s_i-regularand s_i-connected. Moreover, whenever s_i = s_j, G_i and G_j areisomorphic. 1991 Mathematics Subject Classification 05C70.     

Common Multiples of Complete Graphs

A graph H is said to divide a graph G if there exists a setS of subgraphs of G, all isomorphic to H, such that the edgeset of G is partitioned by the edge sets of the subgraphs inS. Thus, a graph G is a common multiple of two graphs if eachof the two graphs divides G. This paper considers common multiples of a complete graph oforder m and a complete graph of order n. The complete graphof order n is denoted K_n. In particular, for all positive integersn, the set of integers q for which there exists a common multipleof K₃ and K_n having precisely q edges is determined. It is shown that there exists a common multiple of K₃ and K_nhaving q edges if and only if q 0 (mod 3), q 0 (mod n2) and (1) q 3 n2 when n 5 (mod 6); (2) q (n + 1) n2 when n is even; (3) q {36, 42, 48} when n = 4. The proof of this result uses a variety of techniques includingthe use of Johnson graphs, Skolem and Langford sequences, andequitable partial Steiner triple systems. 2000 MathematicalSubject Classification: 05C70, 05B30, 05B07.     

On the Genus of a Finite Classical Group

Let G be a finite group acting faithfully and transitively ona set of size m, and let E = {x₁, ..., x_k} be a generatingset for G with x₁x₂...x_k = 1. If x G has cycles of length r₁,..., r_l in its action on , define . Then the genus g = g(G, , E) is defined by 1991 Mathematics Subject Classification 20B25, 20G40,30F99.     

An Algebraic Obstruction to Isomorphism of Markov Shifts with Group Alphabets

Given a compact group G, a standard construction of a Z² Markovshift _G with alphabet G is described. The cardinality of G (ifG is finite) or the topological dimension of G (if G is a torus)is shown to be an invariant of measurable isomorphism for _G.We show that if G is sufficiently non-abelian (for instanceA₅, PSL₂(F₇) or a Suzuki simple group) and H is any abeliangroup with |H| = |G|, then _G and H are not isomorphic. Thusthe cardinality of G is seen to be necessary but not sufficientto determine the measurable structure of _G.     

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.     

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).     

The Glauberman Correspondence and Subgroups of Operator Groups: a Counterexample

Let G and A be finite groups with coprime orders, and supposethat A acts on G by automorphisms. Let (G, A):Irr_A(G)Irr(C_G(A))be the Glauberman–Isaacs correspondence. Let B A andIrr_A(G). We exhibit a counterexample to the conjecture that(G, A) is an irreducible constituent of the restriction of (G,B) to C_G(A). 1991 Mathematics Subject Classification 20C15.     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

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.     

A Sharp Comparison Result Concerning Schrodinger Heat Kernels

Let G₀ and G be the heat kernels of and –V respectively.Under some sharp conditions on V, a proof is given that G(x,t;y,0)/G₀(x,t;y,0)can be bounded from above and below by two positive constants.This largely improves the well-known bounds G(x,t;y,0)/G₀(x,t;y,0) t (0,T]. The result answers an open question of V. Liskevichand Yu. Semenov, in the case where V 0. A sharp global boundin time is also obtained when V 0. 2000 Mathematics SubjectClassification 35K05.