Tarski monster group

In the area of modern algebra known as

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