Tarski group
A Tarski group is an infinite group $G$ such that every non-trivial proper subgroup^{} of $G$ is of prime order.
Tarski groups are also called Tarski monsters, especially in the case when all the proper non-trivial subgroups^{} are of the same order (that is, when the Tarski group is a $p$-group (http://planetmath.org/PGroup4) for some prime $p$).
Alexander Ol’shanskii[1, 2] showed that Tarski groups exist, and that there is a Tarski $p$-group for every prime $p>{10}^{75}$.
From the definition one can easily deduce a number of properties of Tarski groups. For example, every Tarski group is a simple group^{}, it satisfies the minimal condition and the maximal condition, it can be generated by just two elements, it is periodic but not locally finite^{}, and its subgroup lattice (http://planetmath.org/LatticeOfSubgroups) is modular (http://planetmath.org/ModularLattice).
References
- 1 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.
- 2 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.
Title | Tarski group |
---|---|
Canonical name | TarskiGroup |
Date of creation | 2013-03-22 15:46:00 |
Last modified on | 2013-03-22 15:46:00 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 10 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20F50 |
Defines | Tarski monster |