On the symmetric generation of finite groups

Fairbairn, Benjamin Thomas (2009). On the symmetric generation of finite groups. University of Birmingham. Ph.D.

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Abstract

In this thesis we discuss some uses and applications of the techniques in Symmetric generation. In Chapter 1 we introduce the notions of symmetric generation. In Chapter 2 we discuss symmetric presentations defined by symmetric generating sets that are preserved by a group acting on them transitively but imprimitively. In Chapter 3 our attention turns to Coxeter groups. We show how the Coxeter-Moser presentations traditionally associated with the families of finite Coxeter groups of types A $_n$ , D $_n$ and E $_n$ (ie the “simply laced” Coxeter groups) may be interpreted as symmetric presentations and as such may be naturally arrived at by elementary means. In Chapter 4 we classify the irreducible monomial representations of the groups L $_2$ (q) and use these to define symmetric generating sets of various groups.