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 groups of finite max-length with max-length 2. They are thus Artinian groups as well as 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.