Timeline for What property do small primes have that prevent the existence of a Tarski monster?
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| Dec 1, 2019 at 1:44 | comment | added | user6976 | $10^{75}$ is an old bound. The best current is 1003 (odd) proved in Adyan, S. I.; Lysënok, I. G. Groups, all of whose proper subgroups are finite cyclic. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 933–990; translation in Math. USSR-Izv. 39 (1992), no. 2, 905–957. Note that for some exponents, say, powers of 2 Tarski monsters do not exist for simple reasons. | |
| Sep 11, 2018 at 18:44 | comment | added | Wojowu | I would actually suggest editing the question as to reflect that the question is not quite about existence of Tarski monsters per se, but rather for when we can prove the existence (cf Derek's comment). | |
| Sep 8, 2018 at 3:42 | comment | added | Rob | There's a Q&A at Math.SE: Tarski Monster group with prime 3 or 5 with 14 UpVotes. | |
| Sep 8, 2018 at 1:01 | comment | added | zibadawa timmy | @PaulPlummer My limited understanding is that it's established in the following: S I Adyan and I G Lysënok, ON GROUPS ALL OF WHOSE PROPER SUBGROUPS ARE FINITE CYCLIC, Mathematics of the USSR-Izvestiya Volume 39 Number 2. | |
| Sep 8, 2018 at 0:57 | comment | added | YCor | I don't know if there has been any any attempt to show that there no Tarski monster 5-group. If true, it's in principle easier than showing that there's no infinite finitely generated group of exponent 5 (for which there have been many attempts). | |
| Sep 8, 2018 at 0:17 | comment | added | zibadawa timmy | @JohannesHahn No, I think I feel the same way as you at this point. Ben basically has it right: the argument intricately involves small cancellation arguments, the net effect of which tends to create a dizzying series of inequalities you need to hold, and you can force this by just picking the (prime) number to be sufficiently large. So the necessity in the proof isn't anything in particular about the primes, but rather the methods deployed. Especially evident since I understand it has since been proven that p>1000 works. | |
| Sep 7, 2018 at 23:47 | comment | added | Johannes Hahn | @zibadawatimmy There is a construction in Olshanskii's book "Geometry of defining relations in groups". Personally, I find it close to unreadable, but maybe you can work with it. | |
| Sep 7, 2018 at 21:53 | comment | added | zibadawa timmy | I'm having trouble even finding an English copy of Oshanskii's paper that proved their existence (it's behind a paywall I can't get past), so I can't even try to peruse the construction to find where the large primes become necessary. If I knew Russian I'd be fine, but alas...The site-that-shall-not-be-named in particular keeps trying to feed me a paper from astrophysics about the mass-luminosity relationship. In fact almost every paper on this topic appears to have this problem; something odd with IOP's doi's. | |
| Sep 7, 2018 at 21:04 | comment | added | YCor | von Neumann never conjectured "von Neumann's conjecture". | |
| Sep 7, 2018 at 20:34 | comment | added | Benjamin Steinberg | Usually large powers are good for small cancelation arguments | |
| Sep 7, 2018 at 19:39 | comment | added | Derek Holt | Well they cannot exist for $p=3$, because groups of exponent $3$ are easily proved to be locally finite, but I don't believe that they have been proved not to exist for other smaller primes, so we can't talk about reasons for them not existing. I guess the question is rather why the proofs only work for very large primes. | |
| Sep 7, 2018 at 19:33 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |