Talk:Group of Lie type

Definition

I'm concerned that the definition given here is wrong for the generality that is being requested. I don't think PSL_n for example is an algebraic group (scheme). There is SL_n of course, but when you factor out the center, you get PGL_n. But PGL_n(q) is not always PSL_n(q). I think what is described here is a 'finite reductive group'. — Preceding unsigned comment added by 129.67.110.104 (talk) 00:24, 1 December 2011 (UTC)

Also, there is no explicit, and easy to see, definition of what a 'Group of Lie Type' actually is. (This is the reason I was looking this concept up in Wikipedia, otherwise I would write down the definition.) — Preceding unsigned comment added by 128.232.110.146 (talk) 13:26, 8 November 2012 (UTC)

Indeed, I've just looked at the article and can't see a definition anywhere. "A group closely related to the group, G(k), of rational points of a reductive linear algebraic group" isn't really a definition at all. We really need somebody who is familiar with such groups to write a definition for the article. In doing so, WP:MTAA would be something to consider. — Smjg (talk) 11:20, 3 May 2016 (UTC)

@Smjg: I've reworded the intro to better reflect the terminology as it is commonly used. Does it look better? Mark MacD (talk) 05:50, 10 May 2016 (UTC)

Relation to Lie groups

perhaps a note on how these differ from Lie groups? MarSch 14:48, 5 March 2006 (UTC)

AFAIK, a Lie group is a special case of a group of Lie type, with the field being Reals or Complex numbers, in which case the group obtains an extra structure of a smooth manifold. This is certainly true for most examples in Lie group article, but there are some, of which I have no idea (like Lorentz and Poincare) if they can be identifyed as groups of Lie type... --SH 111 07:29, 2 August 2006 (UTC)

Missing information

I was exploring some algebra pages, and I found this one. I don't know much about lie groups, and this page sounded very awkward to me. The text felt pieced together. Two examples: Under Classical Groups the article refers to "this question" -- which question? Also, under Chevalley groups, it refers to "the theory" -- what thoery? So, if anyone has some free time, this would be a good place to do some copyedit :) - grubber 02:14, 17 September 2006 (UTC)

"Bulk"

The article states "Finite groups of Lie type form the bulk of finite simple groups". The word "bulk" here is misleading. There are (countably) infinitely many finite simple groups; in particular there are infinitely many which are not groups of Lie type (e.g. all cyclic groups). A more correct phrase might be "Most of the standard families of finite simple groups are of groups of Lie type"? This is certainly true: any finite simple group falls into one of the following families: "cyclic groups", "alternating groups (n>=5)", "Chevalley groups", "Steinberg groups", "Suzuki-Ree groups", "Sporadic groups" and it is clear to see that of the infinite families, 3 out of 5 of these are families of groups of Lie type. However the sentence as it stands at the moment is not really true. Kidburla (talk) 01:35, 24 May 2008 (UTC)

I agree. Indeed the bulk of finite simple groups are cyclic.

A reasonable measure of bulk would be the limit of the proportion of simple groups of lie type of order less than n amongst all simple groups of order less than n, as n goes to infinity. Roughly speaking the proportion is 1/n, with (n-1)/n being the cyclic groups. Amongst the non-abelian simple groups however, the groups of lie type form the overwhelming majority (in fact, those of type PSL(2,q) form the majority).

I don't think many people count the cyclic groups as groups of Lie type, but they are the fixed points of the algebraic group G_a under standard frobenius maps, so they are particularly boring groups of lie type. Leaving out only the alternating groups and the sporadics makes the claim of "bulk" sound more believable. Perhaps merely adding the word non-abelian would be sufficient? JackSchmidt (talk) 03:47, 24 May 2008 (UTC)

Is this the same Jack Schmidt posting on sci.math? Anyway, I think your measure of bulk is very reasonable, and it certainly seems reasonable that given any n, most of the non-alternating simple groups of order less than n are of Lie type (|PSL(2,q)| viewed as a function of q is essentially polynomial, and so grows a lot more slowly than factorials). I'm not sure if there is a more straight-forward way of putting this across in the article, but for now I think adding the word "non-abelian", as you suggest, will suffice. Kidburla (talk) 20:33, 24 May 2008 (UTC)

Sounds good. If someone later on becomes worried about a citation for this claim, there is a list of nonabelian simple groups of order under 10^largenumber that mentions it has to omit the groups of rank 1 (PSL(2,q)). I post occasionally on sci.math, usually on group theory topics. JackSchmidt (talk) 21:45, 24 May 2008 (UTC)

This is too close to splitting hairs (those who know something will understand anyway, and those who don't will not be served by extra qualifiers). "Bulk" appears to be an excellent non-committal (i.e. intentionally imprecise, yet fairly descriptive) term. I feel that, on the contrary, the mention of Steinberg, Suzuki–Ree, etc in the last line of the lead is already an overkill. Arcfrk (talk) 21:54, 29 May 2008 (UTC)

Dickson

References to all Dickson's papers on the exceptional Lie groups would be useful. 66.130.86.141 (talk) 12:17, 13 February 2010 (UTC)John McKay

Dickson may have also constructed E6 groups over finite fields. Scott Tillinghast, Houston TX (talk) 08:00, 24 February 2011 (UTC)

Graph Automorphisms

Why does the Graph automorphism article just talk in generalities about automorphisms of graphs, and never talk about Lie groups? —Preceding unsigned comment added by 86.202.223.192 (talk) 17:12, 12 December 2010 (UTC)

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