Effective algorithmic approaches to the study of linear groups

Eamonn O'Brien (University of Auckland)

We will consider some of the problems inherent to the study of representations defined over finite fields, and will identify the limitations to existing approaches. We will then report on an on-going research program, which uses the Aschbacher classification of maximal subgroups of linear groups to develop a new paradigm for their study.

Finally, we discuss how to recognize constructively that a given group of 2×2 invertible matrices defined over a finite field defines the special linear group. This is an important base case for a solution to a more general problem.

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