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. A Tarski monster group is necessarily simple. It was shown by Ol'shanskii in 1979 that such groups exist, for p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem.