Groupes et algébres de lie

In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

Fusion Systems in Algebra and Topology

A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

http://emis.matem.unam.mx/journals/EJC/Surveys/ds16.pdf

Dynamic Cage Survey

Takes an exploratory approach to the development of an accountability framework for environmental performance of the Australian Commonwealth public sector. Explains that the aim of the environmental performance accountability framework is for the various stakeholders to understand the actions of Commonwealth public sector organizations and consequences of those actions for ecological systems; to become familiar with the trends and changes in trends in public sector environmental performance; and to place stakeholders in a position to promote change when performance is not acceptable. Examines three interrelated strands of literature. Identifies key institutional stakeholders in the Commonwealth public sector. Synthesizes the literature related to public sector transformation, and reviews the flourishing literature on environmental accounting and reporting, and links it to the concept of environmental accountability. Drawing on this literature, explores interrelationships between three characteristics of environmental performance at the federal level in the Australian public sector: criticality of natural capital; information uncertainty; and regulatory response. Examines environmental accountability dimensions of each of these characteristics. Concludes by making three policy recommendations: first, measurement of environmental performance should focus on criticality of natural capital and informational uncertainty as bounded by the precautionary principle; second, the measures of criticality of natural capital and informational uncertainty should determine the extent of direct accountability to parliament for each public sector organization’s environmental performance; and, finally, a third party attestation of reported information is needed. Also makes some suggestions for extending this exploratory research.

Accountability for environmental performance of the Australian Commonwealth public sector

The diameter of the commuting graph of a finite group with trivial centre

There is no upper bound for the diameter of the commuting graph of a finite group

We verify a conjecture of Bauer Catanese and Grunewald by proving a stronger result concerning quasisimple groups More specifically, we prove that every finite quasisimple group apart from A5 and its cover SL(2,5) is a Beauville group

Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces

We construct a family of finite special 2-groups which have commuting graph of increasing diameter

The commuting graph of a finite soluble group with trivial centre is investigated. It is shown that the diameter of such a graph is at most 8 or the graph is disconnected. Examples of soluble groups with trivial centre and commuting graph of diameter 8 are provided.

Christopher Parker

Papers

The diameter of the commuting graph of a finite group with trivial centre

There is no upper bound for the diameter of the commuting graph of a finite group

Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces

There is no upper bound for the diameter of the commuting graph of a finite group

The commuting graph of a soluble group