Solvable group

In mathematics, a solvable group or soluble group is a group that is formed from by groups whose group operation is commutative.
Relation with the order of the group.
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).