# Tarski 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 $p$ be a fixed prime number. An infinite group $G$ is called a Tarski Monster group for $p$ if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has $p$ elements.

## Properties

• $G$ is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
• $G$ is simple. If $N\trianglelefteq G$ and $U\leq G$ 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 an example of non-amenable groups not containing a free subgroup.
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