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  Explicitly constructing subgroups of exceptional algebraic groups

   School of Mathematics

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  Dr David Craven  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.

Mathematics (25)

Funding Notes

Also open to self-funded students.

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