Maximal cocliques and cohomology in rank one linear groups

Saunders, Jack Phillip ORCID: 0000-0002-6517-0815 (2020). Maximal cocliques and cohomology in rank one linear groups. University of Birmingham. Ph.D.

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Abstract

In this thesis, we investigate certain aspects of \(\mathrm{PSL}_2(q)\). We begin by looking at the generating graph of \(\mathrm{PSL}_2(q)\), a structure which may be used to encode certain information about the group, which was first introduced by Liebeck and Shalev and further investigated by many others. We provide a classification of maximal cocliques (independent sets) in the generating graph of \(\mathrm{PSL}_2(q)\) when \(q\) is a prime and provide a family of examples to show that this result does not directly extend to the prime-power case. After this, we instead investigate the cohomology of finite groups and prove a general result relating the first cohomology of any module to the structure of some fixed module and a generalisation of this result to higher cohomology. We then completely determine the cohomology \(\mathrm{H}^n(G,V)\) and its generalisation, \(\mathrm{Ext}_G^n(V,W)\), for irreducible modules \(V\), \(W\) for \(G = \mathrm{PSL}_2(q)\) for all \(q\) in all non-defining characteristics before doing the same for the Suzuki groups.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):