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Tarski monster group : ウィキペディア英語版 | 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. ==Definition== Let be a fixed prime number. An infinite group is called a Tarski Monster group for if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has elements.
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