For an odd prime $p$, a Tarski monster group is an infinite group $G$ such that every proper, non-trivial subgroup $H < G$ is a cyclic group of order $p$. It is known that for every prime $p > 10^{75}$ a Tarski monster exists. These groups are counter-examples to some famous problems in group theory, including von Neumann's conjecture.

However, that they can only be proven to exist with $p$ so large is very strange to me. What are some properties of small primes that prevent such groups from existing? That is, what is the exact 'law of small numbers' that force such groups to be finite?

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    $\begingroup$ 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. $\endgroup$ – Derek Holt Sep 7 '18 at 19:39
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    $\begingroup$ Usually large powers are good for small cancelation arguments $\endgroup$ – Benjamin Steinberg Sep 7 '18 at 20:34
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    $\begingroup$ von Neumann never conjectured "von Neumann's conjecture". $\endgroup$ – YCor Sep 7 '18 at 21:04
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    $\begingroup$ @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. $\endgroup$ – zibadawa timmy Sep 8 '18 at 0:17
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    $\begingroup$ @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. $\endgroup$ – zibadawa timmy Sep 8 '18 at 1:01

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