Bounds for finite primitive complex linear groups

In 1878, Jordan showed that a finite complex linear group must possess a normal abelian subgroup whose index is bounded by a function of the degree n alone. In this paper, we study primitive groups; when n>12, the optimal bound is (n+1)!, achieved by the symmetric group of degree n+1. We obtain the optimal bounds in smaller degree also. Our proof uses known lower bounds for the degrees of the faithful representations of each quasisimple group, for which the classification of finite simple groups is required. In a subsequent paper [M.J. Collins, On Jordan's theorem for complex linear groups, J. Group Theory 10 (2007) 411–423] we will show that (n+1)! is the optimal bound in general for Jordan's theorem when n⩾71.