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Explicitly constructing subgroups of exceptional algebraic groups


   School of Mathematics

   Applications accepted all year round  Competition Funded PhD Project (Students Worldwide)

About the Project

There is now a complete classification of the Lie primitive (i.e., not contained in a positive-dimensional subgroup) subgroups of the exceptional algebraic groups of types G2, F4 and E6. This is true, in the sense that there is a table in a paper. For G2 there is also a Magma computer program that, given a prime p, returns as 7x7 matrices all Lie primitive subgroups of G2 in characteristic p, written over a minimal subfield of an algebraically closed field. No such program exists for F4 (or the maximal subgroups of the large Ree groups 2F4(q) ) or E6.

This computational mathematics project would produce constructive programs that work fast (the program for G2 finds all Lie primitive subgroups in less than a second for 20-digit primes) giving these subgroups as explicit matrices over finite fields. The final aim is to produce a Magma package where the user can call F4maximals(q) and be given a complete list of all maximal subgroups of the finite group F4(q), a moderately more ambitious goal than just the Lie primitive subgroups. The same would happen for the other series of groups of Lie type.


Funding Notes

Also open to self-funded students.

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