In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group such that every proper subgroup, 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 > 10<sup>75</sup>. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.
Definition
A Tarski group is an infinite group such that all proper subgroups have prime power order. Such a group is then a Tarski monster group if there is a prime <math>p</math> such that every non-trivial proper subgroup has order <math>p</math>.
An extended Tarski group is a group <math>G</math> that has a normal subgroup <math>N</math> whose quotient group <math>G/N</math> is a Tarski group, and any subgroup <math>H</math> is either contained in or contains <math>N</math>.
Properties
As every group of prime order is cyclic, every proper subgroup of a Tarski monster group is cyclic.