Tarski monster group

In mathematics, 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 A. Yu. Ol'shanskii 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.

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.