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Tarski monster group

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(Redirected from Tarski monster) Type of infinite group in group theory This article is about the kind of infinite group known as a Tarski monster group. For the largest of the sporadic finite simple groups, see 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 Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 10. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.

Definition

Let p {\displaystyle p} be a fixed prime number. An infinite group G {\displaystyle G} is called a Tarski monster group for p {\displaystyle p} if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has p {\displaystyle p} elements.

Properties

  • G {\displaystyle G} is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
  • G {\displaystyle G} is simple. If N ⊴ G {\displaystyle N\trianglelefteq G} and U ≤ G {\displaystyle U\leq G} is any subgroup distinct from N {\displaystyle N} the subgroup N U {\displaystyle NU} would have p 2 {\displaystyle p^{2}} elements.
  • The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime p > 10 75 {\displaystyle p>10^{75}} .
  • 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.
  • Ol'shanskiĭ, A. Yu. (1991), Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-1394-6


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