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

Michael Collins (Oxford)

Frank Adams 1,

In 1878, Jordan showed that there is a function \(f:\mathbb{N}\to\mathbb{N}\) such that, if \(G\) is a finite subgroup of \(\operatorname{GL}_n(\mathbb{C})\), 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 \(\operatorname{GL}_n(\mathbb{Q})\) for any given prime \(p\).


I will discuss the history of these problems and how the methods that I used about ten years ago to obtain the optimal bounds for Jordan’s theorem can be adapted to answer a question posed by Serre about an analogue of Minkowski’s bound for the order of a Sylow \(p\)-subgroup of a finite subgroup of \(\operatorname{GL}_n(\mathbb{C})\).


The talk will be at a level suitable for a wide algebraic audience, but hopefully with enough insight to describe the methods to specialists in finite group theory.

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