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Register a new userModular analogues of Jordans theorem for finite linear groups
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Authors
Michael J. Collins
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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.
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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 $mathbb{C}$ by a field of positive characteristic, due to the presence of large unipotent and/or Lie type subgroups. For this reason, a long-standing problem in group and representation theory has been to find the correct analogue of Jordans theorem in characteristic $p>0$. Progress has been made in a number of different directions, most notably by Brauer and Feit in 1966; by Collins in 2008; and by Larsen and Pink in 2011. With a 1968 theorem of Steinberg in mind (which shows that a significant proportion of elements in a simple group of Lie type are unipotent), we prove in this paper that if a finite group $G$ has a faithful representation over a field of characteristic $p$, then a significant proportion of the elements of $G$ must have $p$-power order. We prove similar results for permutation groups, and present a general method for counting $p$-elements in finite groups. All of our results are best possible.
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