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Tarski group
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(Definition)
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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 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 is modular.
- 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.
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"Tarski group" is owned by yark.
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(view preamble)
| Also defines: |
Tarski monster |
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Cross-references: locally finite, periodic, generated by, maximal condition, minimal condition, simple group, number, subgroups, order, prime, proper subgroup, group, infinite
There are 2 references to this entry.
This is version 7 of Tarski group, born on 2006-03-14, modified 2006-03-22.
Object id is 7722, canonical name is TarskiGroup.
Accessed 1973 times total.
Classification:
| AMS MSC: | 20F50 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Periodic groups; locally finite groups) |
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Pending Errata and Addenda
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