# Tarski group

A Tarski group is an infinite group $G$ such that every non-trivial proper subgroup of $G$ is of prime order.

Tarski groups are also called Tarski monsters, especially in the case when all the proper non-trivial subgroups are of the same order (that is, when the Tarski group is a $p$-group (http://planetmath.org/PGroup4) for some prime $p$).

Alexander Ol’shanskii[1, 2] showed that Tarski groups exist, and that there is a Tarski $p$-group for every prime $p>10^{75}$.

From the definition one can easily deduce a number of properties of Tarski groups. For example, every Tarski group is a simple group, it satisfies the minimal condition and the maximal condition, it can be generated by just two elements, it is periodic but not locally finite, and its subgroup lattice (http://planetmath.org/LatticeOfSubgroups) is modular (http://planetmath.org/ModularLattice).

## References

• 1 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.
• 2 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.
Title Tarski group TarskiGroup 2013-03-22 15:46:00 2013-03-22 15:46:00 yark (2760) yark (2760) 10 yark (2760) Definition msc 20F50 Tarski monster