Concept
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. Let be a fixed prime number. An infinite group is called a Tarski monster group for if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has elements. is necessarily finitely generated. In fact it is generated by every two non-commuting elements. is simple. If and is any subgroup distinct from the subgroup would have elements. The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime . Tarski monster groups are an example of non-amenable groups not containing a free subgroup.