Talk:Burnside problem

I don't think the linked article defines the p^∞-group. Cound someone do that here? Orthografer 23:16, 21 October 2006 (UTC)Reply

I've corrected the link: p^∞-groups are also called Prüfer groups (or quasicyclic groups). --Zundark 10:35, 22 October 2006 (UTC)Reply

Defining M to be the intersection of all NORMAL subgroups of finite Index of B(m,n), makes things much easier. See "The Restricted Burnside Problem" by Michael Vaughan-Lee (available at google books)

A recent addition claims that the free Burnside group of rank 2 and exponent 5 has been proven infinite. The only reference is to an arxiv paper, 1105.0847v2, by Heikki Koivupalo and Kazuma Morita. Even setting aside whether such a reference would suffice, I find no paper in the arxiv with Koivupalo as an author. Paper 1105.0847 is written by Kazuma Morita alone, is in the Number Theory section, and is titled Generalization of the theory of Sen in the semi-stable representation case [1]. I cannot find the word "Burnside" in that paper. Unless someone objects, I am reverting that addition. Magidin (talk) 19:19, 31 May 2011 (UTC)Reply

I asked gemeni AI about the problem and this is what it said

The free Burnside group B(m,n) is finite if and only if m = 2 and n is odd. This was proven by Grigory Perelman in 1994.

In particular, B(2, 5) is finite. 2A06:C701:9FC9:8500:5508:2690:A26C:473 (talk) 15:57, 8 October 2024 (UTC)Reply

Stochastic parrots are not reliable sources, especially for mathematics. Please do not add any material gleaned from such sources in the future. The claim is patently false, as the cases of n=2, 4, and 6 are all finite. Magidin (talk) 17:22, 9 October 2024 (UTC)Reply

In the section "Bounded Burnside problem" there is the following claim: "In the case of the odd exponent, all finite subgroups of the free Burnside groups were shown to be cyclic groups." Could someone provide a reference for that claim? In the book "The Burnside Problem and Identities in Groups" by Adjan I could only find Theorem 3.3 (on page 261), which states that this is only true if we assume that the finite subgroup is also abelian. — Preceding unsigned comment added by CGHaus (talk • contribs) 14:32, 9 October 2014 (UTC)Reply

https://books.google.hu/books?id=OaNsCQAAQBAJ&pg=PA21&lpg=PA21&dq=o+yu+schmidt+problem&source=bl&ots=P8b7f_mS-C&sig=lz6NIDDuhEX89ztYt1oEqvHEp2Y&hl=en&sa=X&ved=0CD0Q6AEwCWoVChMIl9_yrpDLxwIVQ-kUCh38rgL3#v=onepage&q=o%20yu%20schmidt%20problem&f=false — Preceding unsigned comment added by 193.224.79.1 (talk) 06:27, 28 August 2015 (UTC)Reply

Should the fact that this question has been solved go in the lead of the article? Currently the opening paragraph makes it seem like an open problem. — Preceding unsigned comment added by Finbob83 (talk • contribs) 19:08, 1 February 2017 (UTC)Reply

Done. — Preceding unsigned comment added by 31.52.252.177 (talk) 13:30, 14 June 2017 (UTC)Reply

"In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, Pyotr Novikov and Sergei Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381."

Is there some compelling reason to obfuscate this encyclopedia entry by using the word "order" in one sentence and "exponent" in the next?

Or do the two words mean something different here?

In either case, I hope someone knowledgeable on the subject will make this clearer. It certainly needs to be.2600:1700:E1C0:F340:3055:C08D:1F4F:1907 (talk) 03:19, 29 September 2018 (UTC)Reply

The words mean different things. Order here is a property of elements (the order of ${\displaystyle x}$  is the smallest ${\displaystyle k}$  such that ${\displaystyle x^{k}=1}$ ); exponent is a property of sets of elements or of the group as a whole (a positive integer ${\displaystyle n}$  such that ${\displaystyle x^{n}=1}$  for all elements in the set/group). Magidin (talk)

I think we should move the "Brief history" section farther down the page, after the "Restricted Burnside problem" section. Right now the history section contains terminology that the reader may not know yet – for example, the exponent of a group is mentioned many times, but not defined until the "Bounded Burnside problem" section. It also mentions periodic groups, while "periodic" isn't defined until the "General Burnside problem" section. Thoughts? – Pillig (talk) 04:38, 19 May 2019 (UTC)Reply

The text asserts, that there were infinite Burnside groups B(m,n) for all integers m>1 and n>8000. The source Лысёнок still has a divisibility condition, i.e. n should be divisible by 16. (see here). Do you have a source for the more general statement in the text?--FerdiBf (talk) 13:06, 4 July 2020 (UTC)Reply

According to this review ([2]) for any ${\displaystyle m\geq 2}$  the Burnside group ${\displaystyle B(m,n)}$  is infinite for odd ${\displaystyle n\geq 665}$  (this is indicated to be contained in the book of Adyan) and for ${\displaystyle n>2^{48}}$  divisible by ${\displaystyle 2^{9}}$  the same holds (by the results of the reviewed article), so Lysënok's result (review here: [3]) seems to be best available for even exponent (I could not find recent references with a better result). Unless somebody finds a reference with a better result the article should be changed to reflect this jraimbau (talk) 12:44, 7 July 2020 (UTC)Reply

just realised that the two results (odd>665 and 16k) together imply the infiniteness of B(2, n) for all integers n >= 16*665=10640. So there is a result for all n large enough. jraimbau (talk) 11:05, 8 July 2020 (UTC)Reply