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 A. 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.

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.

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