text: Historical question concerning Jordan's theorem - MathOverflow
  Historical question concerning Jordan's theorem
  gr.group-theory flag asked May 2 2010 at 19:04Ben Green2,925●18●26
  take proper account of the existence of Chevalley groups in characteristic $p$). link|flag
  answered Apr 27 2011 at 22:37Geoff Robinson9,856●1●21●27 You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
  I know very little about the subject, so I will only give a link. Larsen and Pink have a proof of the generalization of Jordan theorem to characteristic p which do not rely on the CFSG: http://www.math.ethz.ch/~pink/ftp/LP5.pdf (the paper was never published). link|flag
  answered May 3 2010 at 17:35Yiftach Barnea2,310●1●10●18 Yes indeed - and this paper has proved very useful indeed to many of us working in additive combinatorics recently. The paper also contains a proof in characteristic zero, but it is certainly a bit more complicated than the one I described above. – Ben Green May 3 2010 at 17:39
  The paper recently appeard in JAMS ams.org/journals/jams/2011-24-04.
  – Yiftach Barnea Jul 15 2011 at 23:06
  edited May 2 2010 at 22:30
  answered May 2 2010 at 21:23Jim Humphreys26k●37●82 5
  I am not sure if this is an answer to your question, but by now I think Jordan's function $j(n)$ is well understood (you denote it $F(n)$ in your post). In 1984, Weisfeiler found a sharp upper bound in this "post classification" paper. He improved this bounds to $(n+2)!$ in a unpublished manuscript which I myself haven't seen (it is referenced here), with $(n+1)!$ being the lower bound. P.S. I think you are probably right about that argument being "passed over" - the world pre-CFSG was preoccupied with the quest for two decades, and post-CFSG looked nothing like what it was in the good old days. link|flag
  answered May 2 2010 at 21:18Igor Pak8,206●2●17●57 1 Igor, I believe this issue is comprehensively despatched in this paper of M. J. Collins: On Jordan's theorem for complex linear groups. J. Group Theory 10 (2007), no. 4, 411--423. He evaluates $j(n)$ for all $n$ and shows that $(n+1)!$ is the truth for $n \geq 71$ (and not for $n = 70$). But my interest is more in finding the simplest argument that gives some bound. – Ben Green May 2 2010 at 21:28
  I should also point out that by $j(n)$ I think you mean the index of the biggest normal abelian subgroup of $A$; so your $j(n)$ is at most $F(n)!$, but they need not be the same. – Ben Green May 2 2010 at 21:32
  @Ben: The "simplest" argument that gives some bound may be one of those already mentioned, unless somebody finds a really new approach. Jordan's result is impressively general in scope, but the tools relevant to proving it seem quite limited. A more transparent proof would obviously be welcome. But after Frobenius and Schur,
  the published treatments (Curtis-Reiner, Isaacs,...) are basically similar apart from the way they are integrated with other theorems on finite groups and linear groups. Leaving aside the question of best bounds, it's hard to find a fresh approach. – Jim Humphreys May 3 2010 at 17:10
  Jim: I agree completely. I'd still like an actual reference to the argument I sketched above though, if there is one earlier than Tao's blog. – Ben Green May 3 2010 at 17:40