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Algorithms for matrix groups II

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Most first-generation algorithms for matrix groups defined over
finite fields rely on variations of the Schreier-Sims algorithm,
and exploit the action of the group on an set of vectors or subspaces
of the underlying vector space. Hence they face serious practical limitations.

Over the past 25 years, much progress has been achieved on
developing new algorithms to study such groups.
Relying on a generalization of Aschbacher's theorem
about maximal subgroups of classical groups,
they exploit geometry arising from the natural action
of the group on its underlying vector space to
identify useful homomorphisms. Recursive application of these
techniques to image and kernel now essentially allow us to
construct in polynomial time the composition factors
of the linear group. Using the notion of standard generators,
we can realise effective isomorphisms between a final simple group
and its “standard copy”.

In these lectures we will discuss the “composition tree” algorithm
which realises these ideas; and the “soluble radical model” which
exploits them to answer structural questions about the input group.

This talk is part of the Isaac Newton Institute Seminar Series series.

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