nLab solvable group

Definition

Definition

(solvable group)
A group is solvable if it is a finite iterated group extension of an abelian group by abelian groups.

In other words, a group $G$ is solvable if and only if there exists a finite sequence

$$\{ 1\} \;\subset\; G_1 \;\subset\; G_2 \;\subset\; \ldots \;\subset\; G_k \;=\; G \,,$$

in which

1. $G_{j-1}$ is normal in $G_j$

2. the quotient groups $G_j/G_{j-1}$ are abelian.

Solvability can equivalently be expressed in terms of breaking a group down rather than building it up.

Definition

(solvable group)
Given elements $g,h$ of a group $G$, the commutator is $[g,h] := g^{-1} h^{-1} g h$. The commutator subgroup of $G$ is the subgroup of $G$ generated by the commutators $[g,h]$ for all $g, h \in G$. A group is solvable if the series of groups produced by repeatedly taking the commutator subgroup, the derived series, terminates with the trivial group after finitely many steps.

Similar to nilpotent groups, solvable groups have an inductive definition via group extensions.

The class of solvable groups is defined inductively by the following clauses:

1. All abelian groups are solvable.

2. If $1\to G' \to G \to G''\to 1$ is a group extension and both $G'$ and $G''$ are solvable, then $G$ is solvable.

Examples

Example

(Galois groups)
The terminology “solvable groups” comes from elementary Galois theory: every polynomial equation $\phi$ over an integral domain $K$ has a corresponding Galois group $Gal(\phi/K)$, and $Gal(\phi/K)$ is a solvable group if and only if $\phi$ is a solvable equation (meaning that all its solutions in an algebraic closure of $K$ are expressible using the elements of $K$, the field operations, and extraction of roots).

Remark

Informally, Ex. may be thought of as each step in the derived series of the Galois group requiring an additional level of nesting of radicals. If the derived series does not terminate, then the solutions of the polynomial equation cannot be expressed by radicals, no matter how deeply nested.

Example

(nilpotent groups) A nilpotent group is a solvable group given by central group extensions.

Example

(Taketa's theorem)
M-groups are solvable.

(Taketa 1930, Isaacs 1976, thm. 5.12)

In fact:

Example

If a finite group has the property that for each of its complex irreps of dimension $\geq 2$ some multiple of it is induced from an irrep of a proper subgroup, then the group is solvable.

(LMT 2013)

Related concepts

• commutator subgroup

• nilpotent group

References

Textbooks:

• Cornelia Druţu, Michael Kapovich, Chapter 13 of: Geometric group theory, Colloquium Publications 63, AMS 2018 (ISBN:978-1-4704-1104-6, pdf)
• Ian Stewart, Galois Theory, fourth edition. CRC Press, 2015. ISBN:978-1-4822-4582-0. Chapter 14.

Articles:

• L. Goldmakher, “Arnold’s elementary proof of the insolubility of the quintic”. pdf

See also:

• eom: solvable group, Lie group, solvable

• Wikipedia, Solvable group