Solvable group

In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. The concept of solvable (or soluble) groups arose to describe a property shared by the automorphism groups of those polynomials whose roots can be expressed using only radicals (square roots, cube roots, etc., and their sums and products).
A group is called solvable if it has a normal series whose factor groups are all abelian.
For finite groups, an equivalent definition is that a solvable group is a group with a composition series whose factors are all cyclic groups of prime order. This is equivalent because every simple abelian group is cyclic of prime order. The JordanHölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to nth roots (radicals) over some field.
In keeping with George Polya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can figure out", solvable groups are often useful for reducing a conjecture about a complicated group, into a conjecture about a series of groups with simple structure  cyclic groups of prime order.
All abelian groups are solvable  the quotient A/B will always be abelian if both A and B are abelian. But nonabelian groups may or may not be solvable.
A small example of a solvable, nonabelian group is the symmetric group S_{3}. In fact, as the smallest simple nonabelian group is A_{5}, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable.
The group S_{5} is not solvable — it has a composition series {E, A_{5}, S_{5}}; giving factor groups isomorphic to A_{5} and C_{2}; and A_{5} is not abelian. Generalizing this argument, coupled with the fact that A_{n} is a normal, maximal, nonabelian simple subgroup of S_{n} for n > 4, we see that S_{n} is not solvable for n > 4, a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals.
The property of solvability is in some senses inheritable, since:
 If G is solvable, and H is a subgroup of G, then H is solvable.
 If G is solvable, and H is a normal subgroup of G, then G/H is solvable.
 If G is solvable, and there is a homomorphism from G onto H, then H is solvable.
 If H and G/H are solvable, then so is G.
 If G and H are solvable, the direct product G × H is solvable.
Supersolvable group
As a strengthening of solvability, a group G is called supersolvable if it has an invariant normal series whose factors are all cyclic; in other words, if it is solvable with each A_{i} also being a normal subgroup of G, and each A_{i+1}/A_{i} is not just abelian, but also cyclic (possibly of infinite order). Since a normal series has finite length by definition, there are uncountable abelian groups which are not supersolvable; but if we restrict ourselves to finite groups, we can consider the following arrangement of classes of groups:
 cyclic < abelian < nilpotent < supersolvable < solvable < finite grouppl:Grupa rozwiązalna