Morita Equivalent Blocks in Non-Normal Subgroups and p-Radical Blocks in Finite Groups

Let O be a complete discrete valuation ring with unique maximalideal J(O), let K be its quotient field of characteristic 0,and let k be its residue field O/J(O) of prime characteristicp. We fix a finite group G, and we assume that K is big enoughfor G, that is, K contains all the |G|-th roots of unity, where|G| is the order of G. In particular, K and k are both splittingfields for all subgroups of G. Suppose that H is an arbitrarysubgroup of G. Consider blocks (block ideals) A and B of thegroup algebras RG and RH, respectively, where R{O, k}. We considerthe following question: when are A and B Morita equivalent?Actually, we deal with ‘naturally Morita equivalent blocksA and B’, which means that A is isomorphic to a full matrixalgebra of B, as studied by B. Külshammer. However, Külshammerassumes that H is normal in G, and we do not make this assumption,so we get generalisations of the results of Külshammer.Moreover, in the case H is normal in G, we get the same resultsas Külshammer; however, he uses the results of E. C. Dade,and we do not.     

Generalized Blocks of Unipotent Characters in the Finite General Linear Group

In an article of 2003, Külshammer, Olsson, and Robinson defined ?-blocks for the symmetric groups, where ? > 1 is an arbitrary integer, and proved that they satisfy an analogue of the Nakayama Conjecture. Inspired by this work and the definitions of generalized blocks and sections given by the authors, we give in this article a definition of d-sections in the finite general linear group, and construct d-blocks of unipotent characters, where d ≥ 1 is an arbitrary integer. We prove that they satisfy one direction of an analogue of the Nakayama Conjecture, and, in some cases, prove the other direction. We also prove that they satisfy an analogue of Brauer's Second Main Theorem.     

The Loewy Series of the Steinberg-Pim of Finite General Linear Groups

In this paper we calculate the Loewy series of the projectiveindecomposable module of the unipotent block contained in theGelfand–Graev module of the finite general linear groupin the case of non-describing characteristic and Abelian defectgroup. 2000 Mathematics Subject Classification 20C33.     

The Permutation Modules for Finite (Affine) General Linear Groups

The permutation representations of finite general linear and affine groups on the set of vectors of their standard module are studied. The permutation modules over fields and local rings are considered. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 13, Algebra, 2004.     

Transfer in Hochschild Cohomology of Blocks of Finite Groups

We develop the notion of a cohomology ring of blocks of finite groups and study its basic properties by means of transfer maps between the Hochschild cohomology rings of symmetric algebras associated with bounded complexes of finitely generated bimodules which are projective on either side.     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.     

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.     

Nilpotent Injectors in General Linear Groups

In a recent paper, A. Bialostocki (Israel J. Math.41 (1982), 261-273) has defined a nilpotent injector in an arbitrary finite group G to be a maximal nilpotent subgroup of G, containing a subgroup H of G of maximal order satisfying class (H) ≤2. In the present paper, the author determines the nilpotent injectors of GL(n, q) and shows that they form a unique conjugacy class of subgroups of GL(n, q). It is also proved that if n ≠ 2 or n = 2 and q ≠ 9 is not a Fermat prime >3, then the nilpotent injectors of GL(n, q) are the nilpotent subgroups of maximal order.     

Unipotent Conjugacy in General Linear Groups

Let U(n,q) be the group of upper uni-triangular matrices in GL(n,q), the n-dimensional general linear group over the field of q elements. The number of U(n,q)-conjugacy classes in GL(n,q) is, as a function of q, for fixed n, a polynomial in q with integral coefficients.     

Restriction Between Cohomology Algebras of Blocks of Finite Groups

Let k be an algebraically closed field of characteristic p and G be a finite group. Let N be a normal subgroup of G and c be a G-stable block of kN. We shall discuss the cohomology algebra of the block c, defined by M. Linckelmann and, in this case a generalized block cohomology which can be defined using some generalized Brauer pairs, denoted (c, G)-Brauer pairs, which are introduced by R. Kessar and R. Stancu. We also analyze the restriction map between these two cohomology algebras associated to the block c through transfer maps between the Hochschild cohomology algebras of kGc and of the block c.     

On Recognition by Spectrum of Finite Simple Linear Groups over Fields of Characteristic 2

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L _n (2^k) are recognizable by spectrum for n = 2^m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005.     

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.     

Branching Rules for General Linear Groups

In this article we give a branching rule for Harish–Chandra restriction from the general linear group Gl _n (q) to the Levi subgroup Gl _n?1(q) × Gl₁(q) in the case of the unipotent block.     

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence

M. Linckelmann defined the cohomology algebras of blocks of finite groups. This note is an attempt to analyze an inclusion of cohomology algebras of blocks that corresponds under Brauer correspondence through transfer maps between the Hochschild cohomology algebras of the blocks.Presented by Jon Carlson.     

Lie Regular Generators of General Linear Groups

In this article, we introduce the idea of Lie regular elements and study 2 × 2 Lie regular matrices. It is shown that the linear groups GL(2, ?_{2 ⁿ} ), GL(2, ? _{p ⁿ} ), and GL(2, ?_2p ) (where p is an odd prime) can be genrated by Lie regular matrices. Presentations of linear groups GL(2, ?₄), GL(2, ?₆), GL(2, ?₈), and GL(2, ?₁₀) are also given.     

Periodic Groups Saturated with the Linear Groups of Degree 2 and the Unitary Groups of Degree 3 over Finite Fields of Odd Characteristic

Let M denote the set of the simple 3-dimensional unitary groups U₃ and the simple linear groups L₂ over finite fields of odd characteristic.We prove that each periodic group saturated with groups in M is locally finite and isomorphic to either U₃(Q) or L₂(Q) for a suitable locally finite field Q of odd characteristic.     

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence II

Let k be an algebraically closed field of characteristic p. We shall discuss the cohomology algebras of a block ideal B of the group algebra kG of a finite group G and a block ideal C of the block ideal of kH of a subgroup H of G which are in Brauer correspondence and have a common defect group, continuing (Kawai and Sasaki, Algebr Represent Theory 9(5):497–511, 2006). We shall define a (B,C)-bimodule L. The k-dual L ^* induces the transfer map between the Hochschild cohomology algebras of B and C, which restricts to the inclusion map of the cohomology algebras of B into that of C under some condition. Moreover the module L induces a kind of refinement of Green correspondence between indecomposable modules lying in the blocks B and C; the block varieties of modules lying in B and C which are in Green correspondence will also be discussed.