Tarski group

Tarski groups , named after Alfred Tarski , are examined in the mathematical sub-area of group theory; they are infinite groups with a condition on their subgroups . Some authors also speak of Tarski monster groups or, after their discoverer AJ Olschanski , of Olschanski groups.

definition

A group is called a Tarski group if: ${\displaystyle G}$

• ${\displaystyle G}$is infinite ,
• any real, non-trivial subgroup is finite of prime - order .

A group is called an extended Tarski group if there is a normal divisor such that: ${\displaystyle G}$ ${\displaystyle N\subset G}$

• The quotient group is a Tarski group, ${\displaystyle G/N}$
• ${\displaystyle N}$is cyclic of prime power order> 1,
• for each subgroup applies or . ${\displaystyle H\subset G}$ ${\displaystyle H\subset N}$ ${\displaystyle N\subset H}$

Historical remarks

The structure of the subgroup association of a Tarski group

Alfred Tarski had raised the question of whether there are infinite groups whose subgroup association is level 2, that is, it looks like this. The existence of such groups was unclear for a long time, finally Olschanski showed in 1979 that there are p groups of this kind for prime numbers . At the same time further counterexamples were found for the restricted Burnside problem , which asks the question whether finitely generated groups with a finite group exponent have to be finite. Since Tarski groups are created from two elements (see below), one has further counterexamples of the desired kind with them. Furthermore, it follows that there can be and no Tarski groups, because otherwise the Burnside groups or would have to be infinite which is not the case. ${\displaystyle p>10^{75}}$ ${\displaystyle p=2}$ ${\displaystyle p=3}$ ${\displaystyle p}$ ${\displaystyle B(2,2)}$ ${\displaystyle B(2,3)}$

The structure of the subgroup association of an extended Tarski group

Subgroup association

Since two different, real subgroups and one Tarski group each have prime order, their average must be trivial . The subgroup generated by them must match the total group, otherwise it would be of prime order and and would have to contain what led to. Hence the real, non-trivial subgroups of a Tarski group form an anti-chain . ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle U\cap V}$ ${\displaystyle \langle U\cup V\rangle }$ ${\displaystyle \langle U\cup V\rangle }$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle U=\langle U\cup V\rangle =V}$

The structure of the subgroup association of Tarski groups and extended Tarski groups looks like in the adjacent sketches, in particular it is about M groups .

Since Tarski groups are generated from two elements according to the above and therefore extended Tarski groups are finitely generated, they cannot be locally finite . (Groups are locally finite if every finitely generated subgroup is finite.)

Conversely, Tarski groups appear in infinite M groups generated by two elements as follows:

Let there be an M-group and two elements of prime power order. Let the product of these two elements be infinite. Then: ${\displaystyle G}$ ${\displaystyle x,y\in G}$ ${\displaystyle H:=\langle x,y\rangle }$

• Is so is a Tarski group. ${\displaystyle \langle x\rangle \cap \langle y\rangle =\{1\}}$ ${\displaystyle H}$
• Is so is an extended Tarski group. ${\displaystyle \langle x\rangle \cap \langle y\rangle \not =\{1\}}$ ${\displaystyle H}$

Torsion groups

It is clear that Tarski groups are torsion groups , because if a Tarski group is an element , the subgroup generated by is a real subgroup, otherwise it would be cyclic, i.e. isomorphic to , but is not a Tarski group. As a real subgroup of a Tarski group it must be finite, that is, is a torsion group. From this one can easily see that extended Tarski groups are also torsion groups. When describing the subgroup association, it was already stated that it is an M-group. ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle x}$ ${\displaystyle \langle x\rangle =\{x^{n}\mid n\in \mathbb {Z} \}}$ ${\displaystyle G}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \langle x\rangle }$ ${\displaystyle G}$

Conversely, Tarski groups and extended Tarski groups appear as components of such groups according to a sentence by R. Schmidt as follows:

A torsion group is an M-group if and only if it is the direct product of

• Tarski groups,
• extended Tarski groups
• and a local group,

is, so that every two elements from different direct factors have coprime orders.

simplicity

Tarski groups are easy . Namely, let be a non-trivial normal divisor of the Tarski group . Then it is finite and therefore infinite. An element in that is different from the one element has finite order and therefore creates a real, non-trivial subgroup in . Your archetype under the quotient mapping is then a subgroup that really lies between and . This must finally be of prime order and contain a real subgroup. This contradiction shows that there can be no normal divisor, that is, it is simple. ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle G/N}$ ${\displaystyle G/N}$ ${\displaystyle G/N}$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle G}$

Individual evidence

1. ↑ LN Shevrin, AJ Ovsyannikov: Semigroups and their Subsemigroup Lattices , Springer-Verlag, 1996, ISBN 978-94-015-8751-8 , chapter 5.13
2. ^ Roland Schmidt: Subgroup Lattices of Groups , Walter de Gruyter (1994), ISBN 3-11-011213-2 , page 82: Torsion groups with modular subgroup lattices
3. ^ BH Neumann: Some new rumors in group theory , Math. Medley 6 (3), pages 100-103
4. ↑ A. Yu. Olshanskii: Infinite groups with cyclic subgroups , Dokl. Akad. Nauk SSSR, 245: 4 (1979), 785-787
5. ^ Roland Schmidt: Subgroup Lattices of Groups , Walter de Gruyter (1994), ISBN 3-11-011213-2 , Lemma 2.4.17
6. ↑ Ragmar Rudolph: A subgroup set for modular groups , monthly books for mathematics, Volume 94 (1982), pages 149-153
7. ↑ P. Pálfy : Groups and Lattices in Groups St Andrews 2001 in Oxford , London Mathematical Society, Lecture Notes Series 305, Volume II, Cambridge University Press (2003), ISBN 0-521-53740-1 , page 432, Theorem 2.5
8. ^ Roland Schmidt: Subgroup Lattices of Groups , Walter de Gruyter (1994), ISBN 3-11-011213-2 , Theorem 2.4.16
9. ↑ Roland Schmidt: Groups with a modular subgroup association , Arch. Math 46, pages 118-124 (1986)
10. ↑ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , Chapter 14.4, Exercise 1