For a prime $p$, a group $G$ is a Tarski monster, if it is infinite, and every proper subgroup has order $p$. 

If I am correctly informed, then the Tarski monster was defined to demonstrate our poor understanding of infinite groups, because such monsters obviously don't exist, it should be easy to prove that they don't exist, but we cannot prove it. 

Then Olshanskii proved that Tarski monsters do exist for all large primes, and by now many people believe that "large" means something like $p\geq 11$.