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).     

Noetherian Semigroup Algebras

A complete structural characterization of submonoids S of apolycyclic-by-finite group such that the semigroup algebra K[S]over a field K is right noetherian is obtained. It follows thatsuch algebras are also left noetherian. 2000 Mathematics SubjectClassification 16P40, 16S36, 20M25 (primary), 20F22, 20C07,20M10 (secondary).     

A Proof of a Generalized Nakayama Conjecture

In a recent paper, Külshammer, Olsson, and Robinson proveda deep generalization of the Nakayama conjecture for symmetricgroups. We provide a similar but a shorter and relatively elementaryproof of their result. Our method enables us to obtain a moregeneral H-analogue of the Nakayama conjecture where H is a setof positive integers. 2000 Mathematics Subject Classification20C30 (primary), 20C15, 20C20 (secondary).     

Mackey Functors and Control of Fusion

This paper presents an algebraic approach to Mislin's theoremthat control of p-fusion is equivalent to inducing a mod-p cohomologyisomorphism. There are consequences for the cohomology of p-permutationmodules. 2000 Mathematics Subject Classification 20J06 (primary),20C05 (secondary).     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

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.     

Morita Equivalent 3-Blocks of the 3-Dimensional Projective Special Linear Groups

If G is a projective special linear group PSL(3,q) with q 4or 7 (mod 9), then a Sylow 3-subgroup of G is elementary abelianof order 9. We show that the principal 3-blocks of any two suchgroups are Morita equivalent. This result and Okuyama's theoremfor PSL(3,4) prove the Broué conjecture for these blocks.1991 Mathematics Subject Classification: 20C05, 20C20.     

On Subgroups of Finite Index in Positively Finitely Generated Groups

This paper proves that a subgroup of finite index in a positivelyfinitely generated profinite group has maximal subgroup growthat most n^log(n). In particular such a subgroup cannot be free,answering a question by L. Pyber. 2000 Mathematics Subject Classification20E28, 20P05.     

The Tits Alternative for Cat(0) Cubical Complexes

A Tits alternative theorem is proved in this paper for groupsacting on CAT(0) cubical complexes. That is, a proof is givento show that if G is assumed to be a group for which there isa bound on the orders of its finite subgroups, and if G actsproperly on a finite-dimensional CAT(0) cubical complex, theneither G contains a free subgroup of rank 2, or G is finitelygenerated and virtually abelian. In particular, the above conclusionholds for any group G with a free action on a finite-dimensionalCAT(0) cubical complex. 2000 Mathematics Subject Classification20F67, 20E08.     

Locally Finite Finitary Skew Linear Groups

Let V be a vector space over the division ring D of infinitedimension. We study locally finite, primitive groups G of finitarylinear automorphisms of V. We show that the derived group G'of G is infinite, simple, and lies in every non-trivial normalsubgroup of G, and that G' G Aut G'. Moreover if char D =0, then G is either the finitary symmetric group or the alternatinggroup on some infinite set. If D is commutative, that is, ifD is a field, then all these results are known and are the consequenceof the collective work of a number of people: in particular(in alphabetical order) V. V. Belyaev, J. I. Hall, F. Leinen,U. Meierfrankenfeld, R. E. Phillips, O. Puglisi, A. Radfordand quite probably others. 2000 Mathematics Subject Classification:20H25, 20H20, 20F50.     

Bounded Generation and Subgroup Growth

The author proves in this paper that every profinite group Gwith polynomial subgroup growth is boundedly generated; thatis, it is a product of finitely many procyclic subgroups. Thisanswers a question of P. Zalesskii. By contrast, if G is a boundedlygenerated group, then the subgroup growth of G is at most n^clogn.As a byproduct, a short, elementary proof demonstrates thatAut(F_r) (for r 2) and many other related groups are not boundedlygenerated. 2000 Mathematics Subject Classification 20E07 (primary),20E22 (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.     

Heights of Characters in Blocks of p-Solvable Groups

In this paper, it is proved that if B is a Brauer p-block ofa p-solvable group, for some odd prime p, then the height ofany ordinary character in B is at most 2b, where p^b is the largestdegree of the irreducible characters of the defect group ofB. Some other results that relate the heights of characterswith properties of the defect group are obtained. 2000 MathematicsSubject Classification 20C15, 20C20.     

Low-Dimensional Representations of Special Linear Groups in Cross Characteristics

The low-dimensional projective irreducible representations incross characteristics of the projective special linear groupPSL_n(q) are investigated. If n 3 and (n, q) (3,2), (3,4), (4,2), (4,3), all such representationsof the first degree (which is (qⁿ – q)/(q – 1) – with = 0 or 1) and the second degree (which is (qⁿ –1)/(q – 1) come from Weil representations. We show thatthe gap between the second and the third degree is roughly q^2n-4.1991 Mathematics Subject Classification: 20C20, 20C33.     

Harish-Chandra Vertices and Steinberg's Tensor Product Theorems for Finite General Linear Groups

Representations of Hecke and q-Schur algebras are closely relatedto those of finite general linear groups G in non-describingcharacteristics. Such a relationship can be described by certainfunctors. Using these functors, we determine the Harish-Chandravertices and sources of certain indecomposable G-modules. TheGreen correspondence is investigated in this context. As a furtherapplication of our theory, we establish Steinberg's tensor producttheorems for irreducible representations of G in non-describingcharacteristics. 1991 Mathematics Subject Classification: 20C20,20C33, 20G05, 20G40.     

Springer Correspondence for Disconnected Exceptional Groups

The determination of the generalized Springer correspondencefor disconnected reductive groups is completed here by beingachieved in the case of disconnected groups of types D₄ in characteristic3 and E₆ in characteristic 2. It is expected that this correspondencewill provide an important ingredient for the computation ofcharacter values of finite groups of Lie type. Some evidenceis provided in support of the latter statement. 2000 MathematicsSubject Classification 20G05 (primary), 20C33 (secondary).     

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.     

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.     

On Growth of Groups with Few Relators

Let G be a group that has a presentation with n generators andm relators, where m < n, and let S be an arbitrary generatingset for G. Then some subset of n-m elements of S freely generatesa free subgroup of G. This leads to the proof of a conjectureof Gromov on growth of groups. 2000 Mathematics Subject Classification20F05, 20E18.     

Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number S_n G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationx^m=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof S_n G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function S_n G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary).