# 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* > 10^{75}. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.

## Definition

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.

## Properties

- 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.

## References

- A. Yu. Olshanskii, An infinite group with subgroups of prime orders, Math. USSR Izv. 16 (1981), 279–289; translation of Izvestia Akad. Nauk SSSR Ser. Matem. 44 (1980), 309–321.
- A. Yu. Olshanskii, Groups of bounded period with subgroups of prime order, Algebra and Logic 21 (1983), 369–418; translation of Algebra i Logika 21 (1982), 553–618.
- Ol'shanskiĭ, A. Yu. (1991),
*Geometry of defining relations in groups*, Mathematics and its Applications (Soviet Series),**70**, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-1394-6