Tarski monster group explained

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

be a fixed prime number. An infinite group

is called a Tarski monster group for

if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has

elements.

is necessarily finitely generated. In fact it is generated by every two non-commuting elements.

is simple. If

N\trianglelefteqG

and

U\leqG

is any subgroup distinct from

the subgroup

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.

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.