Functors between shape categories

Permutation groups of prime power degree are investigated here through the study of the corresponding group algebra of the set of all functions from the underlying set on which the permutation group acts to a finite field of characteristic p. For the case when the permutation group is of degree p² acting on a set consisting of the direct product of two elementary abelian p-groups, the structure of a minimal permutation module is obtained under certain conditions. The proofs do not depend on the recent classification results of finite simple groups.     

Blocks of minimal dimension

Any block with defect group P of a finite group G with Sylow-p-subgroup S has dimension at least |S|²/|P|; we show that a block which attains this bound is nilpotent, answering a question of G. R. Robinson. Received: 20 November 2006     

On blocks of strongly p-solvable groups

We prove that a block of a finite strongly p-solvable group G with defect group P is Morita equivalent to its corresponding block of N_G(Z(J (P))) via a bimodule with endopermutation source. Received: 7 December 2005     

Inequalities for some blocks of finite groups

We analyze some inequalities for numbers of characters in p-blocks of finite groups and prove them in the case of blocks with a normal defect group, as well as in the case of certain blocks in non-abelian simple groups. Received: 27 October 2005     

Gluing representations via idempotent modules and constructing endotrivial modules

Let G be a finite group and k be a field of characteristic p. We show how to glue Rickard idempotent modules for a pair of open subsets of the cohomology variety along an automorphism for their intersection. The result is an endotrivial module. An interesting aspect of the construction is that we end up constructing finite dimensional endotrivial modules using infinite dimensional Rickard idempotent modules. We prove that this construction produces a subgroup of finite index in the group of endotrivial modules. More generally, we also show how to glue any pair of kG-modules.     

The Fitting submodule

Let H be a finite group, a field and V a finite dimensional H-module. We introduce the Fitting submodule F_V (H), an H submodule of V which has properties similar to the generalized Fitting subgroup of a finite group. Received: 26 August 2005     

On <Emphasis Type="Italic">c</Emphasis>-normal maximal and minimal subgroups of Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups of finite groups

A subgroup H of a finite group G is called c-normal in G if there exists a normal subgroup N of G such that G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups of which every maximal subgroup of its Sylow p-subgroup is c-normal and the class of groups of which some minimal subgroups of its Sylow p-subgroup is c-normal for some prime number p. Some interesting results are obtained and consequently, many known results related to p-nilpotent groups and p-supersolvable groups are generalized.     

The group of endotrivial modules in the normal case

The group of endotrivial modules has recently been determined for a finite group having a normal Sylow p-subgroup. In this paper, we give and compare three different presentations of a torsion-free subgroup of maximal rank of the group of endotrivial modules. Finally, we illustrate the constructions in an example.     

Cartan invariants and defect zero elements

In this note we prove relations between Cartan invariants in characteristc p and p defect zero elements. Exploiting duality arguments in characteristic two we show that the existence of real 2-defect zero elements is equivalent to the existence of odd diagonal entries in the Cartan matrix corresponding to real-valued Brauer characters. In addition, we present an independent and elementary proof of Theorem 3.3 part (iii) in [2]. Received: 9 January 2006     

A note on the p-local rank

In this paper, we give an inductive definition of p-local rank of a p-block in a finite group G with p||G| and show a necessary and sufficient condition for a p-block B such that plr(B) = 2.Received: 10 March 2004     

Fields of values of Fong characters

If G is a p-solvable finite group with a p-complement H and φ ∈ IBr(G), then P. Fong showed that there exists α ∈ Irr(H) such that α^G=Φ_φ. In this note we prove that α can be chosen such that the field of values index divides φ (1)p. Received: 6 May 2005     

On chief factors of finite groups

Let H and K be normal subgroups of a finite group G and let K≤H. If A is a subgroup of G such that AH=AK or A∩H=A∩K, we say that A covers or avoids H/K respectively. The purpose of this paper is to investigate factor groups of a finite group G using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results.     

Parabolic subgroups of positive modality

In this paper we prove a criterion for a parabolic subgroup P of a reductive algebraic group G to be of positive modality, i.e. to act with an infinite number of orbits on the unipotent radical. This condition is simply a lower bound on the length of the descending central series of the radical. We also show that in this situation we can always find a proper P-invariant linear subspace in the Lie algebra of the radical of P which admits an infinite number of P-orbits, unless P is maximal.This research was supported by ARC Grant # A69030627 (chief investigator: Prof. G. Lehrer).     

The influence of minimal subgroups of focal subgroups on the structure of finite groups

A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G=HK and H∩K=1. In this paper, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in N_G(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained.     

On vertices of exterior powers of the natural simple module for the symmetric group in odd characteristic

We determine the vertices of certain exterior powers of the natural simple -module in odd characteristic p and, in particular, of the simple -module for the case n = pw and w ≢ 1 (mod p). Received: 22 February 2007     

Green vertices and sources of simple modules of the symmetric group labelled by hook partitions

We determine the Green vertices and sources of many of the simple modular representations of the finite symmetric group being parametrized by hook partitions. Received: 20 August 2006