Tarski monster group

Short description: Type of infinite group in group theory

This article is about the kind of infinite group known as a Tarski monster group. For the largest of the sporadic finite simple groups, see 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 > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.

## Definition

A Tarski group is an infinite group T such that all proper subgroups have prime power order. Such a group is then a Tarski monster group if there is a prime p such that every non-trivial proper subgroup has order p.[1]

An extended Tarski group is a group G that has a normal subgroup N whose quotient group G/N is a Tarski group, and any subgroup H is either contained in or contains N.[1]

A Tarski Super Monster (or TSM) is an infinite simple group such that all proper subgroups are abelian, and is more generally called a Perfect Tarski Super Monster when the group is perfect instead of simple. There are TSM groups which are not Tarski monsters.[2]

## Properties

As every group of prime order is cyclic, every proper subgroup of a Tarski monster group is cyclic.[1] As a consequence, the intersection of any two different proper subgroups of a Tarski monster group must be the trivial group.[1]

* G is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
* G is simple. If N⊴G and U≤G is any subgroup distinct from N the subgroup NU would have p2 elements.
* The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime p>1075.
* Tarski monster groups are examples of non-amenable groups not containing any free subgroups.

## References

1. ↑ 1.0 1.1 1.2 1.3 Liu, Lisa. "On the Classification of Tarski Monsters". https://math.mit.edu/research/highschool/primes/circle/documents/2021/Liu.pdf.
2. ↑ Herzog, Marcel; Longobardi, Patrizia; Maj, Mercede (October 6, 1998). "On Generalized Dedekind Groups and Tarski Super Monsters". Journal of Algebra 226.

* 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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