Mathematics Colloquium

Bounds for Finite Linear Groups: From Jordan and Minkowski to a Question of Serre

Michael Collins, Emeritus Professor, Mathematics, Oxford University,

In 1878, Jordan showed that there is a function f on the set of natural numbers such that, if G is a finite subgroup of GL(n,C), then G has an abelian normal subgroup whose index is bounded by f(n). In a rather different vein, in 1891 Minkowski obtained a bound on the order of a p-subgroup of GL(n,Q), for any given prime p.

I will discuss the history of these problems and how the methods that I used to obtain the optimal bounds for Jordan's theorem can be adapted to answer a recent question posed by Serre about an analogue of Minkowski's bound for the order of a Sylow p-subgroup of a finite subgroup of GL(n,C).

The talk will be at a general level suitable for a wide audience, including undergraduates who have completed Ma 5, but hopefully with enough insight to describe the methods to specialists.

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For more information, please contact Mathematics Department by phone at 626-395-4335 or by email at [email protected].