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?