Tarski monster group
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In mathematics, 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. Such groups are now known to exist. This was shown by Ol'shanskii in 1979. They are a source of counterexamples in other questions in group theory.