Geoffrey R. Robinson

Local control in fusion systems of p-blocks of finite groups

Abstract If p is an odd prime, b a p -block of a finite group G such that SL (2, p ) is not involved in N G ( Q , e )/ C G ( Q ) for any b -subpair ( Q , e ), then N G ( Z ( J ( P ))) controls b -fusion, where P is a defect group of b . This is a block theoretic analogue of Glaubermans ZJ -Theorem. Several results of general interest about fusion and blocks are also proved.

Blocks with p-power character degrees

Let B be a p-block of a finite group G. If X(1) is a p-power for all X ∈ Irr(B), then B is nilpotent.

Nilpotent Injectors and Conjugacy Classes in Solvable Groups

We provide an upper bound for the order of a nilpotent injector of a finite solvable group with Fitting subgroup of order n . We also show that the same bound is an upper bound for the number of conjugacy classes, provided that the k(G V) -conjecture holds for solvable G all primes dividing n .

Lifting to non-integral idempotents

In the first section of this paper, we illustrate (for a finite group G and the field of fractions K of a suitable complete dvr R) how to produce all central idempotents of Z(KG) from knowledge of the images of Z(RG) modulo certain powers of J(R). In the second section, we outline another approach to lifting elements closely related to idempotents in more general rings, which may be of wider interest.