Towards a nonsolvable Quotient Algorithm

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• Alexander Hulpke (Colorado State University)
• Thursday 30 January 2020, 10:10-11:00
• Seminar Room 1, Newton Institute.

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GRAW02 - Computational and algorithmic methods

algorithms have been a principal tool for the computational
investigation of finitely presented groups as well as for constructing groups.
We describe a method for a nonsolvable quotient algorithm, that extends a
known finite quotient with a module.
Generalizing ideas of the $p$-quotient algorithm, and building on results of
Gaschuetz on the representation module, we construct, for a finite group
$H$, an irreducible module $V$ in characteristic $p$, and a given number of
generators $e$ a covering group of $H$, such that every $e$-generator
extension of $H$ with $V$ must be a quotient thereof. This construction uses
a mix of cohomology (building on rewriting systems) and wreath product methods.
Evaluating relators of a finitely presented group in such a cover of a known
quotient then yields a maximal quotient associated to the cover.
I will describe theory and implementation of such an approach and discuss
the scope of the method.

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

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