Talk:Burnside's lemma

Application to Fermat's Little Theorem
If anyone has looked at the Wikipedia page called "Proofs of Fermat's Little Theorem", there is one which uses "bracelets" to establish the theorem. If you look at a proof of Cauchy's theorem given by Dr. Eyal Goren on page 79 in his course notes for an algebra course given this fall (http://www.math.mcgill.ca/goren/MATH235.2006/CourseNotesMath235.2006.pdf) you can see how the bracelet argument is virtually identical to the first part of the proof. I'm suggesting that under "example applications" we could mention that the CFF can be used to prove Fermat's Little Theorem quite easily, or even just prove it directly as a nice "sample application". feedback?

DavidKawrykow 01:23, 8 May 2007 (UTC)DavidKawrykow

The example might look better if the tabular information was formatted in a proper table. My HTML skills are weak. Is there anyone who knows what they are doing willing and able to give it a try? hawthorn


 * Nice Proof!


 * I hate this particular notation for orbits stabilsers and fixes. Unless you are very familiar with this :stuff you have to constantly look back at the definitions to see which is which and it is so easy to mix :them up. Does anyone else feel this way also? I don't suppose we could engineer a switch to the much more :instantly memorable
 * $$ \mbox{Orb}_G(x) $$, $$ \mbox{Stab}_G(x) $$ and $$ \mbox{Fix}_X(g) $$?
 * hawthorn


 * Excelent idea. In our country is the notation a bit different (also quite confusing, nevertheless I would say a bit more logical) and I confused orbits with fixes from this reason... Finiteautomaton 20:46, 5 January 2009 (UTC)  —Preceding unsigned comment added by 78.99.139.169 (talk)


 * I would support that change. The proof is kind of hard to understand. FelixP 19:16, 27 May 2006 (UTC)


 * This article confuses two signficantly different results, the Burnside lemma (the stated identity, due to Cauchy, as Burnside knew-- he never claimed priority, and the work of Cauchy would have been known to the readers of his textbook, but some later writer apparently didn't know this literature and jumped to an incorrect conclusion), and the Polya enumeration formula (unknown to Polya, this had been previously found by Redfield). The latter explicitly refers to the cycle index of a finite group.  See any good combinatorics text.  For my article on Newton's identities, we need to deconflate these two results.  TIA---CH  (talk) 20:37, 16 August 2005 (UTC)


 * I agree that the proof is hard to understand due to the non-universal notation and therefore needing to click around to other articles for definitions. I expanded the proof to mitigate this problem, briefly including some definitions, but kept the notation since that is what is used elsewhere (e.g. Orbit-stabilizer theorem, Lagrange's theorem). Solitonic (talk) 18:07, 13 April 2014 (UTC)

infinite set
Is there a good reason to allow this finite group to act on an infinite set? There are always an infinite number of orbits and the identity fixes infinitely many points, so the result is trivial then. It doesn't really hurt but maybe it reduces the understandability. McKay 07:43, 17 June 2006 (UTC)

As far as I know, the assertion is fairly easy to prove true for infinite cardinals. I have added this (stated appropriately). —Preceding unsigned comment added by 173.3.85.24 (talk) 18:57, 3 June 2010 (UTC)

which consequently is a natural number or +∞
The average of natural numbers is not always a natural number. For example avg(2,3)=2.5. Therefore it does not follow that the number of orbits should be a natural number or +∞ from the summand being a natural number or +∞. Crasshopper (talk) 05:34, 12 January 2012 (UTC)
 * The number of orbits is a natural number of +∞ because it is counting something. It is neat that it is also equal to the average, since as you observe, one would not have expected the average to even be a natural number.  For example, S3 acting on 3 points has fixed point sets of sizes 3, 0, 0, 1, 1, 1, and the average size is (3+0+0+1+1+1)/6 = 1; coincidentally (or not, as Burnside discovered) this is the number of orbits of S3 on 3 points. JackSchmidt (talk) 16:10, 12 January 2012 (UTC)
 * Does any book actually state the Burnside formula for an infinite set? Some1Redirects4You (talk) 04:45, 30 April 2015 (UTC)

Proof order
The proof is such that the right-hand side $$\frac{1}{|G|}\sum_{g \in G}|X^g|$$ is manipulated until it equals the left-hand side $$|X/G|$$. Might it make more sense to start with the left-hand side, the value that we are trying to compute, and manipulate it to a formula, the right-hand side? Having a path from a question to an answer is the more natural direction for me. Starting with an answer and then exploring to find a question it resolves is a bit backwards in my book. Leegrc (talk) 14:14, 3 July 2014 (UTC)

odd claim
"William Burnside stated and proved this lemma, attributing it to Frobenius 1887, in his 1897 book on finite groups. But, even prior to Frobenius, the formula was known to Cauchy in 1845. In fact, the lemma was apparently so well known that Burnside simply omitted to attribute it to Cauchy." -- That makes no sense. If it was so well known that he didn't bother attributing it to Cauchy, why did he attribute it to Frobenius? McKay (talk) 03:40, 28 May 2015 (UTC)


 * You're right, that doesn't make much sense. I notice that the edit that introduced the claim that "Burnside simply omitted to attribute it to Cauchy" (this one) says that Burnside proved this lemma without attribution in his 1897 book. So we really need to check the book to see whether Burnside does give any attribution for it. (It's possible that the attribution was added in the second edition, which would explain the confusion.) --Zundark (talk) 08:09, 28 May 2015 (UTC)


 * Actually it is not impossible. Someone today could cite the lemma to anything convenient, say a reference book, rather than to Cauchy. Probably Neumann's article has enough information to clarify the wording. McKay (talk) 05:53, 30 May 2015 (UTC)