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Short presentations of finite groups

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Finding ‘nice & compact’ presentations of various groups has been a subject of much research for more than a century. The Coxeter presentation of the symmetric groups and the Steinberg presentation of groups of Lie type are such. In response to conjectures of Babai and Szemeredi on one hand (motivated by questions in computational group theory) and of Mann on the other hand (motivated by questions on subgroup growth) we show that all non-abelian finite simple groups (with the possible exception of Ree groups) have presentations which are small (bounded number of relations) and short (w.r.t the length of the relations). This is surprising as the simple abelian groups – the cyclic groups of prime order – do not have such presentations! We will describe the motivations and results, a cohomological application (proving a conjecture of Holt) and some connections with discrete subgroups of Lie groups and topology.

The talk will be suitable for undergraduates with basic knowledge of group theory.

The talk is based on a series of papers with Bob Guralnick, Bill Kantor and Martin Kassabov.

1. Presentations of finite simple groups: profinite and cohomological approaches, Groups, Geometry and Dynamics 1 (2007) 469-523.
2. Presentations of finite simple groups: a quantitative approach, J. of the AMS 21 (2008) 711-774.
3. Presentations of finite simple groups: a computational approach, J. of the Euro. Math. Soc. 2 (2011) 391- 458.

This talk is part of the Mordell Lectures series.

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