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

#### Michael Collins (Oxford)

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 $$\operatorname{GL}_n(\mathbb{C})$$.