# Tarski group

From Groupprops

## Definition

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

Tarski groups do not exist for all (for instance, there is no Tarski group for ). However, Tarski groups exist for all large enough primes . Specifically, for all , there is a Tarski group for . In fact, there are infinitely many pairwise non-isomorphic Tarski monsters for each such fixed .

## Significance

- The existence of a Tarski group for a prime provides a negative answer to the Burnside problem for that prime .
- 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.