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I'm interested in Jordan's theorem which (after applying the unitary trick) states that any finite subgroup of $U_n(\mathbb{C})$ has an abelian subgroup of index $F(n)$, a function depending only on $n$ and not on the finite group.

The "standard" reference for this theorem is, I think, the book of Curtis and Reiner (Representation theory of finite groups and associative algebras, Chapter 36), and I think the argument there is due to Frobenius and Schur. But on Tao's blog

http://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/

there's a slight variant on this argument which I personally find a little more natural.

Here's a very brief sketch. We work by induction on n. Let A be our finite subgroup of $U_n(\mathbb{C})$. Look at the intersection $A'$ of $A$ with a small ball about the identity; this must constitute a reasonable fraction of $A$. Divide into two cases. Case 1: everything in $A'$ is a multiple of $I_n$. Then we are done simply by looking at the subgroup of $A$ generated by $A'$. Case 2: there is an element $\gamma \in A'$ which is not a multiple of $I_n$. Take the nearest such element to $I_n$. Then one may argue that for any $x \in A'$ the commutator $[\gamma,x]$ is both nearer to $I_n$ than $\gamma$ and not a nontrivial multiple of $I_n$: hence it must equal $I_n$ and so $\gamma$ is centralised by the whole of $A'$. However the centraliser of an element such as $\gamma$ is a product of $U_{n_i}(\mathbb{C})$'s with $n_i < n$, and that means we can proceed by induction.

I asked Terry about this and he said someone had sketched the "choose an element closest to the identity" part of the argument and this had seemed the most natural way to conclude.

Anyway, I'd be interested in any comments people have on this or on proofs of Jordan's theorem in general (for example on Jordan's original proof). Can one find the above argument somewhere in the literature? Or was it passed over because it doesn't give especially good bounds on the index of the abelian subgroup?

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4 Answers

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 He improved this bounds to $(n+2)!$ in a unpublished manuscript which I myself haven't seen (it is referenced here), with $(n+1)!$ being the lower bound.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.

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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 '10 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 '10 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 '10 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 '10 at 17:40
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Some useful references are given below, readily available online (with access to JSTOR). Weisfeiler's short paper was published just before his disappearance while hiking in Chile in January 1985, but some details have circulated in manuscript form. (Igor Pak has just now posted an overlapping account.) The original source is the paper by C. Jordan, J. Reine Angew. Math. 84 (1878), 89–215].

MR0200350 (34 #246) 20.65 Brauer, Richard; Feit,Walter, An analogue of Jordan’s theorem in characteristic p. Ann. of Math. (2) 84 1966 119–131.

MR758425 (85j:20041) 20G20 (20D05) Weisfeiler, Boris (1-PAS), Post-classification version of Jordan’s theorem on finite linear groups. Proc. Nat. Acad. Sci.U.S.A. 81 (1984), no. 16, Phys. Sci., 5278–5279.

ADDED: The specific reference Igor mentions is B. Weisfeiler, On the size and structure of finite linear groups, preprint. My understanding is that several such typed manuscripts kept at Penn State by Don James lack references and the like, but Boris's sister Olga has made an effort to get them published online in completed format. See the current status of her effort at the page linked by Igor which she maintains (end of list): http://weisfeiler.com/boris/papers/papers.html, and contact her if you have any useful information from other sources.

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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).

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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 '10 at 17:39
    
The paper recently appeard in JAMS ams.org/journals/jams/2011-24-04. –  Yiftach Barnea Jul 15 '11 at 23:06
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I gave a proof along those lines in my imaginatively titled paper "On linear Groups", JOURNAL OF ALGEBRA 131, 527-534 (1990), in an issue dedicated to Walter Feit. I regarded this proof as being in the tradition of classical proofs of Jordan, Frobenius, and Blichfeldt. One point I wanted to make was that the the two apparently different methods of proof found in standard texts were essentially equivalent: one method was to prove that elements of finite complex linear groups which have their eigenvalues sufficiently close together commute with all their conjugates (Blichfeldt, Frobenius are generally credited with proofs on these lines). The other method was to show that elements sufficiently close to the identity commute with each other( see, for example, the proof credited to Schur in the 1962 book of Curtis and Reiner). For unitary matrices, heving eigenvalues sufficiently close to each other on $S^{1}$ is equivalent to being close to a scalar in the operator norm on $M_{n}(\mathbb{C}).$ The upshot is that in a finite subgoup $G$ of $U_{n}(\mathbb{C})$, the elements which lie at distance less than $\frac{1}{2}$ (in the operator norm) from any scalar generate an Abelian normal subgroup $A$ of $G$, and that elements in different cosets of $A$ in $G$ are more than $\frac{1}{2}$ apart. Hence $[G:A]$ is bounded by the number of open balls of radius $\frac{1}{4}$ which cover the closed unit ball of $M_{n}(\mathbb{C})$. As already noted in some of the remarks, the issue of a general proof for the existence of a bound, which is (by modern standards) a relatively easy compactness argument is quite a different issue from knowing what the best possible bound is in each dimension. The papers of Weisfeiler and Collins give (via the Classification of Finite Simple Groups) close to optimal/respectively optimal bounds over the complex field, and analogous results in characteristic $p$ (where one must take proper account of the existence of Chevalley groups in characteristic $p$).

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