Polycyclic group

Polycyclic groups are special groups considered in the mathematical branch of group theory . They are made up of cyclical groups .

definition

A group is called polycyclic if it is a finite chain ${\displaystyle G}$

${\displaystyle \{1\}=G_{0}\vartriangleleft G_{1}\vartriangleleft \ldots \vartriangleleft G_{n}=G}$

so that each factor group is cyclical . As usual, the symbol stands for "is normal divisor in". ${\displaystyle G_{i+1}/G_{i}}$ ${\displaystyle \vartriangleleft }$

Examples

• Every cyclic group is polycyclic.
• Any finite, solvable group is polycyclic, because a resolution can be refined to one with simple factors, and simple Abelian groups are cyclic.
• Every indissoluble group is polycyclic, in particular finitely generated nilpotent groups are polycyclic.
• The infinite dihedral group is polycyclic, but not nilpotent.

properties

• Subgroups, homomorphic images and extensions of polycyclic groups are again polycyclic.
• Polycyclic groups meet the maximum condition for subgroups, i.e. every non-empty set of subgroups has a maximum element .

Proof: This is clear for cyclic groups and the maximum condition continues on extensions.

• Every subgroup of a polycyclic group is finitely generated, because that is equivalent to the maximum condition.
• Every polycyclic group is residual finite , that is, for every element different from 1 there is a normal subgroup with a finite index that does not contain the element.
• The Frattini group of a polycyclic group is nilpotent .
• If G is a group that contains a polycyclic subgroup with a finite index, then the group ring with respect to a field K is Noetherian . ${\displaystyle K[G]}$

Equivalent characterizations

• A group is polycyclic if and only if it can be resolved and the maximum condition is met.
• A group G is polycyclic if and only if there is a series of normal subdivisions such that all factors are either a finitely generated free Abelian group or a finite elementary Abelian group . ${\displaystyle \{1\}=G_{0}\vartriangleleft G_{1}\vartriangleleft \ldots \vartriangleleft G_{n}=G}$ ${\displaystyle G_{i}\vartriangleleft G}$ ${\displaystyle G_{i+1}/G_{i}}$
• The polycyclic groups are, apart from isomorphism, precisely the solvable subgroups of the integral general linear group . ${\displaystyle \mathrm {GL} _{n}(\mathbb {Z} )}$

Anatoly Malzew already proved in 1951 that resolvable subgroups of the are polycyclic . The proof of the reversal assumed by Philip Hall was made by Louis Auslander in 1967 , the proof was considerably simplified by Richard Swan . ${\displaystyle \mathrm {GL} _{n}(\mathbb {Z} )}$

Deer length

The cyclic series in the definition of the polycyclic group is not clearly defined, as the simple example shows. But the number of factors that are too isomorphic does not depend on the cyclic series. This number is called the Hirsch length of the polycyclic group named after KA Hirsch . ${\displaystyle \{1\}=G_{0}\vartriangleleft G_{1}\vartriangleleft \ldots \vartriangleleft G_{n}=G}$ ${\displaystyle \mathbb {Z} _{6}\cong \mathbb {Z} _{3}\oplus \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} }$

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , page 54
2. ↑ Peter J. Hilton: Nilpotent groups and nilpotent spaces , Lecture Notes in Mathematics, Volume 1053 (1981), Definition 3.19
3. ^ Louis H. Rowen: Ring Theory II , Academic Press (1988), according to Definition 8.2.1
4. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , sentence 3.1.6
5. ↑ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , sentence 5.4.17
6. ^ Louis H. Rowen: Ring Theory II , Academic Press (1988), according to Definition 8.2.1
7. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , sentence 5.4.12
8. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , sentence 5.4.14
9. ^ Daniel Segal: Polycyclic Groups , Cambridge University Press (2005), ISBN 978-0-521-02394-8 , Chapter 5
10. ^ AI Malcev: On some classes of infinite solvable groups , Mat. Sb. 28 (70) (1951), pp. 567-588; Amer. Math. Soc. Transl. (2) 2 (1956), pp. 1-22
11. ↑ L. Auslandser: On a Problem of Philip Hall , Annals of Mathematics (1967), Volume 86, No. 1, pp. 112-116
12. ^ R. Swan: Representations of Polycyclic Groups , Proceedings of the American Mathematical Society (1967), Volume 18, pages 573-574, see here
13. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , sentence 5.4.13