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Towards a nonsolvable Quotient Algorithm

Presented by: 
Alexander Hulpke
Thursday 30th January 2020 - 10:10 to 11:00
INI Seminar Room 1

[This is joint work with Heiko Dietrich from Monash U.]
Quotient 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.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons