User talk:JackSchmidt/Archives/2010/03

Minimal (and maximal) Sylow intersections
Hi Jack, I was wondering whether it might be worthwhile to create a Wikipedia article on minimal Sylow intersections; in particular, there are interesting results due to J. S. Brodkey, T. J. Laffey, and M. Herzog on the subject. Essentially, the results are based on the fact that if S is an abelian Sylow p-subgroup of a finite group G, then the normal core of S in G (i.e. the p-core of G) is equal to the intersection of S with just one other Sylow p-subgroup T ($$S\cap T=\mathrm{Core}_G(S)$$).

Do these results collectively warrant an article? I notice that you have done a lot of important work on finite group theory in Wikipedia, and thus you probably have a better sense than I do regarding which concepts/results require Wikipedia articles. We might also include maximal Sylow intersections in the context of the result that a group of order $$p^aq$$ is simple if and only if it is of prime order (of course, this can be proved by applying the famous result due to Burnside on groups of order $$p^aq^b$$, but a simpler proof using maximal Sylow intersections also exists). Thanks! PS T  02:57, 25 February 2010 (UTC)

On a similar note, I noticed a while ago that our article on the Sylow theorems requires significant improvement (as do other articles on finite groups in Wikipedia), but that you are doing an excellent job in expanding and fixing them. I would just like to say thanks for all the work you have done; I am sure that many other users here appreciate it as well. PS T  03:04, 25 February 2010 (UTC)

Hi again, I created an article on the Three subgroups lemma, and a corresponding redirect Hall-Witt identity. If you have the time, could you please have a look at them? There may be specific problems with the article that do not exist in other Wikipedia articles on finite group theory (but I am sure that there are no factual errors). Of course, there are also a couple of interesting facts that are not present in the article. For instance, the neat (yet simple) bound: If d and c are respectively the derived length and nilpotence class of a group, then $$d<1+\log_2(c+1)$$. Thanks! PS T  05:32, 3 March 2010 (UTC)


 * You might consider working on GroupProps instead. A very useful resource to have is roughly a "union textbook" that contains all results from all textbooks (on a single subject area) with links, references, and proofs.  Wikipedia is not really striving for that.  A wikipedia article has to assert notability, should provide historical context, and should balance various points of view.  These constraints requiring omitting more material than they allow to add.  GroupProps is closer.  It is basically the attempt to provide a unifying framework for each piece of group theory the administrator is exposed to, combined with an attempt to explain that group theory back to a larger audience.  I suspect that your goals are similar: here is an interesting bit of mathematics, I want to sharpen my understanding of it and share it with the world by writing it up.  This can produce a few wikipedia articles, but generally will produce too many AfD candidates.  GroupProps may be a much better place to contain the write-ups, and is almost certainly a better place for an enthusiastic writer.
 * I don't think sylow intersections have enough secondary sources for an article. Three subgroups is basically ok, but I suspect it will only be of limited value *on wikipedia* because it is basically a proof technique.  However, it is an incredibly useful proof technique and should be available in some sort of wiki style source.  Frattini argument for whatever reason gets a lot of press, so maybe it is notable, but you can see it is basically only mentioned in 2 out of however many million articles there are on enwiki.  It strikes me as borderline already, and three subgroups is even more tenuous. JackSchmidt (talk) 18:29, 3 March 2010 (UTC)

Sorry and thanks
Thanks for having corrected the erroneous statement about the acyclicity property of the binary icosahedral group. I thought this homotopic fibre X of BG -> BG+ for G = A5 is also a K(pi,1), ridiculous ! And thanks for the useful reference. Sergeraert (talk) 09:31, 25 March 2010 (UTC)