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In 1878, Jordan 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. 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.
This is a nearly complete manuscript left behind by Boris Weisfeiler before his disappearance during a hiking trip in Chile in 1985. It is posted on a request from the authors sister, Olga Weisfeiler.
In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces $mathbb{C}$
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