Tarski monster group

In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p &gt; 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.

Definition
Let $$p$$ be a fixed prime number. An infinite group $$G$$ is called a Tarski monster group for $$p$$ if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has $$p$$ elements.

Properties

 * $$G$$ is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
 * $$G$$ is simple. If $$N\trianglelefteq G$$ and $$U\leq G$$ is any subgroup distinct from $$N$$ the subgroup $$NU$$ would have $$p^2$$ elements.
 * The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime $$p>10^{75}$$.
 * Tarski monster groups are examples of non-amenable groups not containing any free subgroups.