Tarski group

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Let p be a prime number. A Tarski group (also called Tarsi monster) for the prime p is an infinite group in which every proper nontrivial subgroup is a group of order p.

Tarski groups do not exist for all p (for instance, there is no Tarski group for p = 2). However, Tarski groups exist for all large enough primes p. Specifically, for all p > 10^{75}, there is a Tarski group for p. In fact, there are infinitely many pairwise non-isomorphic Tarski monsters for each such fixed p.