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.     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

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.     

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).     

Normal Subgroups of Groups Which Split Over The Infinite Cyclic Group

Let G be either a free product with amalgamation A*_CB or anHNN group A*_C, where all normal subgroups of C are finitelygenerated. Suppose that both A and B have no non-trivial finitelygenerated normal subgroups of infinite indices. We show thatif G contains a finitely generated normal subgroup N which intersectsA or B non-trivially but is not contained in C, then the indexof N in G is finite. 1991 Mathematics Subject Classification20E06.     

Modular Branching Rules and the Mullineux Map for Hecke Algebras of Type A

We prove the quantum version - for Hecke algebras H A_n of typeA at roots of unity - of Kleshchev's modular branching rulefor symmetric groups. This result describes the socle of therestriction of an irreducible H A_n-module to the subalgebraH A_n–1. As a consequence, we describe the quantum versionof the Mullineux involution describing the irreducible moduleobtained on twisting an irreducible module with the sign representation.1991 Mathematics Subject Classification: 20C05, 20G05.     

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.     

The Markov Oscillation Problem in Discrete Time

Consider a discrete-time regenerative phenomenon with associatedrenewal sequence u_n. General results for the supremum of u_m+1,..., u_m+n are developed for those renewal sequences {u_n} forwhich the first m + 1 elements match those of a fixed renewalsequence {v_n}, that is, u₀ = v₀, ..., u_m = v_m. A series of associatedlemmas are developed in the process.     

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.     

Supplementary Note on 'Rational Invariants of Certain Orthogonal and Unitary Groups'

Given a field k and a finite group G acting on the rationalfunction field k(X₁, ..., X_n) as a group of k-automorphisms,an important Noether's problem asks whether the invariant subfield [forumal] is purely transcendental over k. 1991 Mathematics Subject Classification12F20, 20G40.     

The Symmetric Genus of the Mathieu Groups

The (symmetric) genus of a finite group may be defined as thesmallest genus of those closed orientable surfaces on whichG acts faithfully as a group of automorphisms. In this paperthe genus of each of the five Mathieu groups M₁₁, M₁₂, M₂₂,M₂₃ and M₂₄ is determined, with the help of some computer calculationsand a little-known of Ree on permutations.     

On Perfect Isometries for Tame Blocks

Any 2-block of a finite group G with a quaternion defect groupQ₈ is Morita equivalent to the corresponding block of the centraliserH of the unique involution of Q₈ in G; this answers positivelyan earlier question raised by M. Broué. 2000 MathematicsSubject Classification 20C20.     

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 Mathieu Group M11 and the Modular Curve X(11)

In this paper, we prove that the modular curve X(11) over afield of characteristic 3 admits the Mathieu group M₁₁ as anautomorphism group. We also examine some aspects of the geometryof the curve X(11) in characteristic 3. In particular, we showthat every point of the curve is a point of inflection, thecurve has 110 hyperflexes and there are no inflectional trianglesand 11232 inflectional pentagons, of which 144 are self-conjugate.The hyperflexes correspond to the supersingular elliptic curves.We comment on the relationship of Ward's quadrilinear invariantfor M₁₂ to our work and announce for the first time the equationsfor Klein's A-curve of level 11. We also comment on the relationof our work to some unpublished work of Bott and Tate. 1991Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10,14N10, 20B25, 20C34.     

Symmetric Groups as Products of Abelian Subgroups

A proof is given that the full symmetric group over any infiniteset is the product of finitely many Abelian subgroups. In fact,289 subgroups suffice. Sharp bounds are also obtained on theminimal number k, such that the finite symmetric group S_n isthe product of k Abelian subgroups. Using this, S_n is provedto be the product of 72n^1/2(log n)^3/2 cyclic subgroups. 2000Mathematics Subject Classification 20B30, 20D40.     

The Betti Numbers of the Free Loop Space of a Connected Sum

Let M₁ and M₂ be two simply connected closed manifolds of thesame dimension. It is proved that (1) if k is a coefficient field such that neither M₁ nor M₂has the same cohomology as a sphere, then the sequence (b_k)_k1of Betti numbers of the free loop space on M₁ #M₂ is unbounded; (2) if, moreover, the cohomology H*(M₁;k) is not generated asalgebra by only one element, then the sequence (b_k)_k1 has anexponential growth. Thanks to theorems of Gromoll and Meyer and of Gromov, thisimplies, in case 1, that there exist infinitely many closedgeodesics on M₁#M₂ for each Riemannian metric, and, in case2, that for a generic metric, the number of closed geodesicsof length t grows exponentially with t.     

The Characterization of the Regularity of the Jacobian Determinant in the Framework of Bessel Potential Spaces on Domains

Let 2 m n. The paper gives necessary and sufficient conditionson the parameters s1, s2, ..., sm, p1, p2, ..., pm such thatthe Jacobian determinant extends to a bounded operator fromHs1p1 x Hs2p2 x ... x Hsmpm into S'. Here all spaces are definedon Rn or on domains Rn. In almost all cases the regularity ofthe Jacobian determinant is calculated exactly.     

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 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 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).