Conditions for a Schunck class to be a formation

A Schunck class H is determined by the class X of primitives contained in H. We give necessary and sufficient conditions on X for H to be a saturated formation.Comment: 6 pages. Version 2 clarifies solubility conditions assumed in section

Too Many Probabilities: Statistical Evidence of Tort Causation

Medical scientific testimony is often expressed in terms of two different probabilities: 1. The increased probability of harm if a person is exposed, for example, to a toxin. 2. The observed relationship is an artifact of the experimental method. This article demonstrates that neither probability, taken alone or together, measures whether the preponderance of the evidence test is met

Character clusters for Lie algebra modules over a field of non-zero characteristic

For a Lie algebra L over an algebraically closed field of non-zero characteristic, every finite-dimensional L-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Clusters are used to generalise the construction of induced modules.Comment: 8 page

Blocks of representations of Lie algebras

In the theory of finite groups, the irreducible representations of G over a field F are classified into blocks based on a direct decompositions of the group algebra FG. This gives a natural decomposition of FG-modules into direct summands, each summand having all its composition factors belonging to a single block. This block decomposition is the finest natural decomposition of the FG-modules. In this paper, a classification of the irreducible representations of a finite dimensional Lie algebra L into blocks is defined, giving the finest natural direct decomposition of L-modules. This classification is investigated for supersoluble algebras.Comment: This paper has been withdrawn by the author due to an error in the proof of Lemma 4.22. It is possible for the extension to spli