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

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

An extended Tarski group is a group ${\displaystyle G}$ that has a normal subgroup ${\displaystyle N}$ whose quotient group ${\displaystyle G/N}$ is a Tarski group, and any subgroup ${\displaystyle H}$ is either contained in or contains ${\displaystyle N}$.^[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]

• ${\displaystyle G}$ 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.