On Jordan's theorem for complex linear groups

In 1878, Jordan showed that a finite subgroup of GL(n, ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n; for n ⩾ 71, it is (n + 1)!, afforded by the symmetric group Sn+1. We prove a 'replacement theorem' that enables us to study linear groups by breaking them down into individual primitive constituents and we give detailed information about the structure of the groups that achieve the optimal bounds, for every degree n.