Tarski monster group

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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 Alexander 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.

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Definition

A Tarski group is an infinite group such that all proper subgroups have prime power order. Such a group is then a Tarski monster group if there is a prime such that every non-trivial proper subgroup has order . [1]

An extended Tarski group is a group that has a normal subgroup whose quotient group is a Tarski group, and any subgroup is either contained in or contains . [1]

A Tarski Super Monster (or TSM) is an infinite simple group such that all proper subgroups are abelian, and is more generally called a Perfect Tarski Super Monster when the group is perfect instead of simple. There are TSM groups which are not Tarski monsters. [2]

Properties

As every group of prime order is cyclic, every proper subgroup of a Tarski monster group is cyclic. [1] As a consequence, the intersection of any two different proper subgroups of a Tarski monster group must be the trivial group. [1]

References

  1. 1 2 3 4 Liu, Lisa. "On the Classification of Tarski Monsters" (PDF).
  2. Herzog, Marcel; Longobardi, Patrizia; Maj, Mercede (October 6, 1998). "On Generalized Dedekind Groups and Tarski Super Monsters". Journal of Algebra. 226.