Tarski monster group

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

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

Properties

• ${\displaystyle G}$  is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
• ${\displaystyle G}$  is simple. If ${\displaystyle N\trianglelefteq G}$  and ${\displaystyle U\leq G}$  is any subgroup distinct from ${\displaystyle N}$  the subgroup ${\displaystyle NU}$  would have ${\displaystyle p^{2}}$  elements.
• The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime ${\displaystyle p>10^{75}}$ .
• 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), vol. 70, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-1394-6