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