# Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups" https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf contain a complete list of isomorphisms" among the various finite simple Chevalley groups (Th. 37 on pp. 108--109). Unfortunately, the proofs are omitted. I am looking for a reference where the completeness of the list is proven.

• In the original types Yale lecture notes, that Theorem 37 occurs on "pp. 200-201", not "pp. 108--109". – nfdc23 Jun 23 '16 at 16:30
• Thanks for pointing out. I've made a reference to a version that was (re)typed in LaTeX. – Yuri Zarhin Jun 24 '16 at 8:44
• I think the completeness of the list is not really that hard -- if you just compare orders and use some basic number theory results like Zsigmondy's theorem, you quickly reduce to just a few possible isomorphisms. Then the job is to construct isomorphisms, where they exist -- most of these are done in Kleidman and Liebeck (at least between classical groups). There are a couple of places where these isomorphisms don't exist -- $L_3(4) \not\cong L_4(2)$ and $Sp_{2n}(q)\not\cong O_{2n+1}(q)$ for $q$ odd -- but these facts are not hard. – Nick Gill Jun 24 '16 at 9:09
• For a reference: The statement you want is Theorem 2.2.10 of Gorenstein-Lyons-Solomon Volume 3. They don't give a proof, but the reference they give (which I can't check because I don't have a copy) is to result 3-3 of Gorenstein-Lyons "The local structure of finite groups of characteristic 2 type" Mem. Amer. Math. Soc. 276 (1983). Actually their result is more general -- it deals with all finite groups of Lie type. – Nick Gill Jun 24 '16 at 9:18

I've tracked down, I think, the best references although I don't have access to them. A description of the history of this question is in Wilhelm Magnus' preface to the Dover edition of Dickson's Linear groups:

In a later paper Dieudonné settled one of the fundamental questions which Dickson had left unanswered by showing that Dickson's list of isomorphisms between the simple groups discussed in his book is complete. Finally, in 1955, E. Artin, in two astonishingly short papers, demonstrated that the whole table for the orders of finite simple groups and the isomorphisms between them can be derived systematically, with discussion of only a very few separate cases.

The references mentioned by Magnus are below.

Jean Dieudonné, MR 45125 On the automorphisms of the classical groups. With a supplement by Loo-Keng Hua, Mem. Amer. Math. Soc., 1951 (1951), no. 2, vi+122.

Emil Artin, MR 70642 The orders of the linear groups, Comm. Pure Appl. Math. 8 (1955), 355--365.

Emil Artin, MR 73601 The orders of the classical simple groups, Comm. Pure Appl. Math. 8 (1955), 455--472.

Note that these references predate the discovery of some of the finite simple groups (e.g. Suzuki's) so they won't deal with all of the finite groups of Lie type. But, given you only want the Chevalley groups, you should be fine.

• By the way, given Dickson's remarks above about "fundamental questions", perhaps I was hasty in saying that proving the completeness of the list is not that hard... I've never actually worked through the proof about the coincidence of orders between the various groups, so it's possible that the number theory is a little more involved than I had imagined... – Nick Gill Jun 24 '16 at 10:26
• Yes, it's hard for me to believe that Artin published two (even one) trivial papers :-) – Yuri Zarhin Jun 24 '16 at 13:25
• @Nick: Note that the remarks you attribute to Dickson were actually made by Wilhelm Magnus in his 1958 preface to Dickson's book (I got acquainted with Magnus while at the Courant Institute, whose house journal in which Artin's papers appeared was then edited by Natascha Artin). By the way, these Artin papers are reprinted in the 1965 volume of his collected papers edited by Lang and Tate. In them he compares in detail the orders of the known simple groups and sorts out which are isomorphic. (One complication is that non-isomorphic groups sometimes have the same order.) – Jim Humphreys Jun 24 '16 at 17:59
• @JimHumphreys, Thanks for your remark -- I hadn't realised that Magnus wrote that preface -- I shall edit the answer accordingly. – Nick Gill Jun 27 '16 at 8:50

In a remark after Thm. 37, Steinberg does cite a paper of Dieudonné in the Can. J. Math. The year "1949" is obviously a typo. It should be "1954" (vol. 6) and refers to Dieudonné's paper "Les isomorphismes exceptionnels entre les groupes classiques finis".

• Thanks for the extra reference. Note that Dieudonne's 1954 paper does rely heavily on his 1951 AMS Memoir treating both automorphisms and isomorphisms, while in both cases he studies just the classical groups (first over arbitrary fields, not only finite). The finite simple groups of exceptional Lie types $E - G$ turn out to have no unexpected isomorphisms, as seen for example in Artin's more detailed study of the group orders and their arithmetic. Probably there is no single source giving all details, but the most important sources have been listed. – Jim Humphreys Jun 27 '16 at 17:37