Talk:Burnside's lemma

message: 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)
thread_title:  infinite set 

message: I think the Neumann paper cited should help with the history section.  The exclamation points recently removed from the article appear to be lifted (to or) from MathWorld.  I don't see any evidence of banging in the literature, but it is not something that is easy to search for, nor is it something that is likely to make it past my selective reality filter.
Without the bangs, I think the article is now basically sourced.  There still remain some nuances that are not sourced, but they are not important to the article (and so could be re-nuanced to sound less like a fan page). JackSchmidt (talk) 14:12, 23 July 2008 (UTC)
thread_title:  Sources 

message: 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)
replies: text: 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)
replies: text:  Does any book actually state the Burnside formula for an infinite set? Some1Redirects4You (talk) 04:45, 30 April 2015 (UTC)
thread_title:  which consequently is a natural number or +∞ 

message: 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)
thread_title:  Proof order 

message: "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)
replies: text: 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)
replies: text:  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)
thread_title:  odd claim