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 such that every proper subgroup, 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.

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 <math>p</math> such that every non-trivial proper subgroup has order <math>p</math>.<ref name="Liu">Template:Cite web</ref>

An extended Tarski group is a group <math>G</math> that has a normal subgroup <math>N</math> whose quotient group <math>G/N</math> is a Tarski group, and any subgroup <math>H</math> is either contained in or contains <math>N</math>.<ref name="Liu" />

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.<ref name="Herzog">Template:Cite journal</ref>

Properties

As every group of prime order is cyclic, every proper subgroup of a Tarski monster group is cyclic.<ref name="Liu" /> As a consequence, the intersection of any two different proper subgroups of a Tarski monster group must be the trivial group.<ref name="Liu" />

• Every Tarski monster group is finitely generated. In fact it is generated by every two non-commuting elements.
• If <math>G</math> is a Tarski monster group, then <math>G</math> is simple. If <math>N\trianglelefteq G</math> and <math>U\leq G</math> is any subgroup distinct from <math>N</math> the subgroup <math>NU</math> would have <math>p^2</math> elements.
• The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime <math>p>10^{75}</math>.
• Tarski monster groups are examples of non-amenable groups not containing any free subgroups.

References

• A. Yu. Olshanskii, An infinite group with subgroups of prime orders, Math. USSR Izv. 16 (1981), 279–289; translation of Izvestia Akad. Nauk SSSR Ser. Matem. 44 (1980), 309–321.
• A. Yu. Olshanskii, Groups of bounded period with subgroups of prime order, Algebra and Logic 21 (1983), 369–418; translation of Algebra i Logika 21 (1982), 553–618.
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