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 $p$.

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 $N$.

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.

Properties

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

• $G$ is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
• $G$ is simple. If $N⊴G$ and $U\le G$ 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 examples of non-amenable groups not containing any free subgroups.

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