Insoluble group

Insoluble group is a term from the mathematical branch of group theory . It is a tightening of the dissolvability of a group .

definition

A group is called solvable , if there are normal divisors with ${\displaystyle G}$ ${\displaystyle G_{i}\vartriangleleft G}$

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

so that all factor groups are cyclical . ${\displaystyle G_{i+1}/G_{i}}$

The main difference to solvability is that we not only require a normal factor to be in , in order to be able to form the factor groups, but also make the stronger requirement that they are even normal factors in . Super-dissolvability is therefore a stronger term than dissolvability. ${\displaystyle G_{i}}$ ${\displaystyle G_{i+1}}$ ${\displaystyle G_{i}}$ ${\displaystyle G}$

Examples

• In a trivial way, every cyclic group can be resolved. Thus the groups and are solvable, as well as finite direct sums of such. ${\displaystyle \mathbb {Z} _{n}}$ ${\displaystyle \mathbb {Z} }$
• Finally generated nilpotent groups can be resolved.
• The symmetrical group S ₃ is solvable but not nilpotent, because

${\displaystyle \{(1)\}\vartriangleleft \{(1),(123),(132)\}\vartriangleleft S_{3}}$

evidently fulfills the definition, but since the group has a trivial center it cannot be nilpotent. ${\displaystyle S_{3}}$

• The infinite dihedral group is solvable but not nilpotent.
• The alternating group A ₄ can be resolved but not over-resolved.

properties

• Super-resolvable groups can be resolved, as was already noted for the definition.
• Insoluble groups are polycyclic .
• Groups that can be resolved satisfy the maximum condition , i.e. every non-empty set of subgroups contains a maximum subgroup. From this it follows that every subgroup is finitely generated. In particular, groups that can be resolved are always finitely generated.
• The defining series of normal subdivisions of an unsolvable group is not clearly determined. By means of suitable operations, one can even go over to a series whose factors are arranged as follows: First, all factors are isomorphic with an odd prime number p , and that in descending order, then all are isomorphic factors and finally all are isomorphic factors. ${\displaystyle \{1\}=G_{0}\vartriangleleft G_{1}\vartriangleleft \ldots \vartriangleleft G_{n}=G}$ ${\displaystyle G_{i+1}/G_{i}}$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} _{2}}$
• If solvable, the fitting subgroup is nilpotent and the factor group is finite and Abelian. ${\displaystyle G}$ ${\displaystyle \mathrm {Fit} (G)}$ ${\displaystyle G/\mathrm {Fit} (G)}$

Inheritance properties

• Subgroups and homomorphic images of dissolvable groups are dissolvable again.
• The converse does not apply, the class of super-resolvable groups is not closed with regard to extensions . The alternating group contains a normal divisor that is isomorphic to Klein's group of four . Then and are dissolvable, but themselves are not dissolvable. ${\displaystyle A_{4}}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle A_{4}/V\cong \mathbb {Z} _{3}}$ ${\displaystyle A_{4}}$
• Certain extensions, however, can be resolved: If a group has a cyclic normal divider so that it can be resolved, then it can be resolved. ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle G/N}$ ${\displaystyle G}$
• Finite direct sums of insoluble groups are again insoluble.
• Infinite direct sums are usually not solvable. So it can not be resolved, because this group does not meet the maximum condition. ${\displaystyle \oplus _{n\in \mathbb {N} }\mathbb {Z} _{2}}$

Finite groups

For finite groups there are some equivalent characterizations for which the following terms are required. denote the Frattini group of the group . A maximal chain in is understood to be a chain of subgroups, so that each maximal subgroup in is for , the number n is the length of this chain. ${\displaystyle \Phi (G)}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \{1\}=M_{0}<M_{1}<\ldots <M_{n}=G}$ ${\displaystyle M_{i}}$ ${\displaystyle M_{i+1}}$ ${\displaystyle 0\leq i<n}$

For a finite group are equivalent: ${\displaystyle G}$

• ${\displaystyle G}$ is dissolvable.
• ( B. Huppert ) Every maximal subgroup has a prime number as an index .
• ${\displaystyle G/\Phi (G)}$ is dissolvable.
• ( K. Iwasawa ) Every two maximal chains in have the same length. ${\displaystyle G}$

The implications hold for finite groups

cyclic     abelian     nilpotent     super-     dissolvable dissolvable. ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$

The above example shows that for infinite groups from abelian it does not necessarily follow super-resolvable. ${\displaystyle \oplus _{n\in \mathbb {N} }\mathbb {Z} _{2}}$

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.4.6. (ii)
2. ^ John C. Lennox: Theory of Infinite Soluble Groups , Clarendon Press (2004), ISBN 978-0-191-52315-1 , page 15
3. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.4.8.
4. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.4.10.
5. ^ WR Scott: Group Theory , Dover Publications (2010), ISBN 978-0-486-65377-8 , sentence 7.2.4
6. ^ WR Scott: Group Theory , Dover Publications (2010), ISBN 978-0-486-65377-8 , sentence 7.2.14
7. ^ WR Scott: Group Theory , Dover Publications (2010), ISBN 978-0-486-65377-8 , sentence 7.2.5
8. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 9.4.4.
9. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 9.4.5.
10. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 10.3.5.