Tarski group

Definition
Let $$p$$ be a prime number. A Tarski group (also called Tarsi monster) for the prime $$p$$ is an infinite group in which every proper nontrivial subgroup is a group of order $$p$$.

Tarski groups do not exist for all $$p$$ (for instance, there is no Tarski group for $$p = 2$$). However, Tarski groups exist for all large enough primes $$p$$. Specifically, for all $$p > 10^{75}$$, there is a Tarski group for $$p$$. In fact, there are infinitely many pairwise non-isomorphic Tarski monsters for each such fixed $$p$$.

Significance

 * The existence of a Tarski group for a prime $$p$$ provides a negative answer to the Burnside problem for that prime $$p$$.
 * Tarski groups give examples of infinite simple non-abelian p-groups, in sharp contrast to the finite case where all p-groups are nilpotent.
 * Tarski groups are Noetherian groups: any Tarski group is a 2-generated group and all its proper subgroups are cyclic. They can be used to illustrate that Noetherian not implies finitely presented and also that Noetherian not implies solvable.