Tarski group

A Tarski group is an infinite group $G$such that every non-trivial proper subgroup of $G$ is of prime order.

Tarski groups are also called Tarski monsters,especially in the case whenall the proper non-trivial subgroups are of the same order(that is, when the Tarski group isa $p$-group (http://planetmath.org/PGroup4) for some prime $p$).

Alexander Ol’shanskii[1, 2] showed that Tarski groups exist,and that there is a Tarski $p$-group for every prime $p>{10}^{75}$.

From the definition one can easily deducea number of properties of Tarski groups.For example,every Tarski group is a simple group,it satisfies the minimal condition and the maximal condition,it can be generated by just two elements,it is periodic but not locally finite,and its subgroup lattice (http://planetmath.org/LatticeOfSubgroups) is modular (http://planetmath.org/ModularLattice).