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Is there any large $p$ for which it is proven that an infinite group with all non-trivial subgroups cyclical of order $p$ doesn't exist? And (if there is), which is the largest such $p$ ? All I could find is the $10^{75}$ upper bound, but I admit I didn"t look very far...

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Aren't there small primes p > 700 for which the finitely generated infinite Burnside groups exist? Or are you looking for p even and nonprime? Gerhard "Ask Me About System Design" Paseman, 2012.10.17 –  Gerhard Paseman Oct 18 '12 at 5:35
    
Huh? p must be prime, as we are looking for "every subgroup" –  Feldmann Denis Oct 18 '12 at 7:29
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I suspect that the largest prime for which this it is known that no Tarski Monster exists is 3 (but I might be wrong)! –  Derek Holt Oct 18 '12 at 9:12
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The largest known prime for which existence of Tarski monster is not known is $997$, see Adyan, S. I.; Lysënok, I. G. Groups, all of whose proper subgroups are finite cyclic. Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 933--990; translation in Math. USSR-Izv. 39 (1992), no. 2, 905–957. There are currently no methods of proving that for a given prime $p$ a Tarski monster does not exist except for showing that all finitely generated groups of exponent $p$ are finite. This gives hope that the bound $997$ can be lowered to $665$ (or below $300$ assuming the recent Adian's announcement is correct).

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