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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 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.

Contents

Definition

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

  • G is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
  • G is simple. If N ⊴ G and U ≤ G is any subgroup distinct from N the subgroup N U would have 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 .

    • References

    Tarski monster group Wikipedia
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