## Abstract

In 1878, Jordan [*C. Jordan*, Mémoire sur les equations différentielle linéaire à intégrale algébrique, J. reine angew. Math. **84** (1878), 89–215.] showed that a finite subgroup of *GL*(*n*, ℂ) contains an abelian normal subgroup whose index is bounded by a function of *n* alone. Previously, the author has given precise bounds [*M. J. Collins*, On Jordan's theorem for complex linear groups, J. Group Th. **10** (2007), 411–423.]. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic ℓ. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic ℓ, and we show that generically the bound is similar to that in characteristic 0—being (*n* + 1)!, or (*n* + 2)! when ℓ divides *n* + 2—given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees *n* and every characteristic ℓ.

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