# Sylow theorem meta question

This is in relation to http://meta.math.stackexchange.com/questions/1868/list-of-generalizations-of-common-questions : I noticed that there have been quite a few questions (recently and otherwise) which to the trained eye immediately says "Apply Sylow-1, then apply Sylow-2, then Sylow-3, then chug."

It is somewhat redundant (in my opinion) to have so many questions whose answers all boil down to standard manipulations plus a computational step involving factoring some number; this question is to ask whether other users feel the same.

• Should we create an "application of Sylow theorems" generalisation and just point users toward it in the future?

• Another possibility would be to create a (sylow-theorem) tag so it would be easier for users to locate those questions (as a model/template for problem solving, I suppose).

Other ideas, suggestions, comments on why this is a bad idea, are welcome.

Per Alex Becker's request in the comment, here's a sample of stuff you get if you search for "group order" on the main site.

• Can you link to some of these questions? I am rather bad at recognizing how to solve them, and would relish the examples. – Alex Becker May 9 '12 at 7:43
• @Alex: considering the audience of this website, my kneejerk reaction to seeing a question asking about structure of group of order $N$ where $N$ is a reasonably small positive integer (such as existence of certain subgroup of certain order, the non-existence of normal subgroups of certain order, whether the group is simple etc.) is that they are almost certainly homework/exercise questions (based on the arbitrariness of the formulation) after a lecture on Sylow theorems have been given. There are, of course, exceptions where more elementary (Cauchy or Lagrange theorems) can be used... – Willie Wong May 9 '12 at 7:55
• ... as well as problems that require (or can be solved more simply by) slightly higher powered machinery. But as you can see from the list I just added there is a definite trend to the "shape" of those questions. – Willie Wong May 9 '12 at 7:57
• An example of an exception is this recent question math.stackexchange.com/questions/142827/… which needs a bit more input than just direct computation. – Willie Wong May 9 '12 at 8:00
• Thank you, those should serve as instructive review for me. – Alex Becker May 9 '12 at 8:45
• I think the "Applications of Sylow's Theorems" generalisation is a very good idea. If for any reason people object to that, then having a sylow-theorem tag might at least help. – Tara B May 9 '12 at 11:16
• Dear Willie, I think that a Sylow-theorems tag would be a good idea. Regards, – Matt E May 9 '12 at 11:35
• I've seen a website, which I can't find at the moment, where one enters a natural number (below some bound, maybe 1000) and the site automatically provides a proof using the Sylow theorems that there are no simple groups of that order, or a classification of the simple groups of that order. – Zev Chonoles May 9 '12 at 16:58
• Do you have in mind a certain question that could be promoted to a catch-all reference? Otherwise I'm willing to try and write something in the space below — I think it would be an interesting diversion. – Dylan Moreland May 9 '12 at 20:10
• @Dylan unfortunately no. The closest I can find is this answer. I wish I still have my undergraduate algebra notes: John Conway gave an algorithm (along with wonderful remarks on why doing it that way is the best) for doing problems like this. So your effort will be very much appreciated. – Willie Wong May 10 '12 at 7:42
• @WillieWong I hope you can dig that up! It brings up an interesting point: without an algorithm, how do you generalize all of the examples given? I suppose the thing to do is talk about the various techniques available (the three main theorems, counting elements of certain orders, embedding into symmetric groups, semidirect products, etc.) and display them in some fundamental examples (order $pq$, $p^2q$, $p^2$, etc.) Is this what we're after? – Dylan Moreland May 10 '12 at 23:57
• @Dylan: I don't have the notes anymore :( but from my vague recollection it is just a step by step instruction for problems of the type "classifying all finite groups of order $m$", which of course starts by finding all the $p$ subgroups, then counting conjugacy classes and ascertain the normal ones, then ruling out impossible combinations by some elementary number theory, and then writing down what the group can be. – Willie Wong May 14 '12 at 7:15