Solvable group

Definition
Solvable is also called soluble by some people.

Equivalent definitions in tabular format
The length of the derived series, and the smallest possible length of a series for any of the other equivalent definitions, is termed the derived length or solvable length of the group.

Extreme examples

 * The trivial group is solvable.
 * Symmetric group:S3 is the smallest solvable non-abelian group.

Formalisms
The group property of being solvable can be obtained in either of these equivalent ways:


 * By applying the poly operator to the group property of being abelian
 * By applying the finite normal series operator to the group property of being abelian
 * By applying the finite characteristic series operator to the group property of being abelian

Note that all these three operators have the same effect in the case of abelian groups, though in general they may not have.

The testing problem
The problem of testing whether a group is solvable or not reduces to the problem of computing its derived series. This can be done when the group is described by means of a generating set, if the normal closure algorithm can be implemented.

To determine whether a group is solvable or not, we cna use the following GAP command:

IsSolvableGroup(group);

where group may be a definition of the group or a name for a group previously defined.

Study of this notion
The class 20F16 is used for the general theory of solvable groups, while the class 20D10 (coming under 20D which is for finite groups) focusses on finite solvable groups.

Also closely related is 20F19: Generalizations of nilpotent and solvable groups.