Solvable group

In mathematics, a solvable group or soluble group is a group which has a subnormal series with abelian factors.

Relation with the order of the group

See the main article: Order (group theory)

The Feit-Thompson theorem says that every group of odd order is solvable.

Every finite group of order 60 or greater, every abelian group, and every subgroup of a solvable group is solvable.

Applications

The definition of solvable groups show why the polynomial equations of degree greater than 5 are (in general) not solvable using finite arithmetic operations (Galois theory).