Tarski group
A Tarski group is an infinite group
such that every non-trivial proper subgroup
of 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 -group (http://planetmath.org/PGroup4) for some prime ).
Alexander Ol’shanskii[1, 2] showed that Tarski groups exist, and that there is a Tarski -group for every prime .
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 |