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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 > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.
Definition

Let p {\displaystyle p} p be a fixed prime number. An infinite group G {\displaystyle G} G is called a Tarski Monster group for p {\displaystyle p} p if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has p {\displaystyle p} p elements.
Properties

G is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
G is simple. If \( N\trianglelefteq G \) and \( U\leq G \) is any subgroup distinct from N the subgroup NU would have \( p^{2} \) elements.
The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime \( p>10^{{75}} \).
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

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