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" > 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 formula_1 be a fixed prime number. An infinite group formula_2 is called a Tarski Monster group for formula_1 if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has formula_1 elements.