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 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[edit]

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

Properties[edit]

  • G is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
  • G is simple. If N\trianglelefteq G and U\leq G is any subgroup distinct from N the subgroup NU would have p^2 elements.
  • The construction of Ol'shanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime p>10^{75}.

References[edit]

  • 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) 70, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-1394-6