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Independent Sets in Some Classical Groups of Dimension Three

Keen, Philip James (2012)
Ph.D. thesis, University of Birmingham.

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Abstract

Given a finite group G, an independent set S in G is a set where no element of S can be written as a word in the other elements of S. A minimax set is an independent generating set for G of largest size in G. This thesis seeks to find a good upper bound for the size of minimax sets in SL(3,q) for odd q. In preparation for this, the sizes of independent sets in SO(3,q) and SU(3,q) are also investigated for odd q.

In each of the cases G = SO(3,q) or SU(3,q), q odd, it is shown that if S is an independent set in G, then either |S| has a particular upper bound, or <S> stabilises some sub-structure of the underlying vector space V. These results are then used to help gain upper bounds for minimax sets in SL(3,q).

Further results are shown for finite groups which contain normal, abelian subgroups. These are then used to obtain the size of minimax sets in finite Coxeter groups of types Bn and Dn.

Type of Work:Ph.D. thesis.
Supervisor(s):Hoffman, Cornelius
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:3288
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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