Tarski monster group

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. 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.

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.


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