Tarski group
From Groupprops
(Redirected from Tarski monster)
Definition
Let be a prime number. A Tarski group (also called Tarsi monster) for the prime
is an infinite group in which every proper nontrivial subgroup is a group of order
.
Tarski groups do not exist for all (for instance, there is no Tarski group for
). However, Tarski groups exist for all large enough primes
. Specifically, for all
, there is a Tarski group for
. In fact, there are infinitely many pairwise non-isomorphic Tarski monsters for each such fixed
.
Significance
- The existence of a Tarski group for a prime
provides a negative answer to the Burnside problem for that prime
.
- Tarski groups give examples of infinite simple non-abelian p-groups, in sharp contrast to the finite case where all p-groups are nilpotent.
- Tarski groups are groups of finite max-length with max-length 2. They are thus Artinian groups as well as Noetherian groups: any Tarski group is a 2-generated group and all its proper subgroups are cyclic. They can be used to illustrate that Noetherian not implies finitely presented and also that Noetherian not implies solvable.