More generally, in keeping with Polya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can't figure out", solvable groups can often be used to reduce a conjecture about a complicated group, into a conjecture about a series of groups with simple structure  cyclic groups of prime order.
Let E be the trivial subgroup; then a normal series of a group G is a finite sequence of subgroups, E = A_{1}, A_{2}, ..., A_{i}, ..., A_{n1}, A_{n} = G, where each A_{i} is a normal subgroup of A_{i+1}. There is no requirement that A_{i} be a normal subgroup of G (a series with this additional property is called an invariant series); nor is there any requirement that A_{i} be maximal in A_{i+1}.
A series with the additional property that A_{i} ≠ A_{i+1} for all i is called a normal series without repetition; equivalently, each A_{i} is a proper normal subgroup of A_{i+1}.
If we require that each A_{i} be a maximal, proper, normal subgroup of A_{i+1}, it then follows that the factor group A_{i+1} / A_{i} will be simple in each case. This gives the following definition: a composition series of a group is a normal series, without repetition, where the factors A_{i+1} / A_{i} are all simple.
There are no additional subgroups which can be "inserted" into a composition series; and it can be seen that, if a composition series exists for a group G, then any normal series of G can be refined to a composition series, informally, by inserting subgroups into the series up to maximality. Every finite group has a composition series; but not every infinite group has one (for example, the additive group of integers (Z, +) has no composition series).
In general, a group will have multiple, different composition series. For example, the cyclic group C_{12} has {E, C_{2}, C_{6}, C_{12}}, {E, C_{2}, C_{4}, C_{12}}, and {E, C_{3}, C_{6}, C_{12}} as different composition series. However, the result of the JordanHolder Theorem[?] is that any two composition series of a group are equivalent, in the sense that the sequence of factor groups in each series are the same, up to rearrangement of their order in the sequence A_{i+1} / A_{i}. In the above example, the factor groups are isomorphic to {C_{2}, C_{3}, C_{2}}, {C_{2}, C_{2}, C_{3}}, and {C_{3}, C_{2}, C_{2}}, respectively.
Finally  a group is called solvable if it has a normal series whose factor groups are all abelian.
For finite groups, it is equivalent (and useful) to require that a solvable group have a composition series whose factors are all cyclic of prime order (as every simple, abelian group must be cyclic of prime order). The JordanHolder 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.
Certainly, any abelian group will be solvable  the quotient A/B will always be abelian if both A and B are abelian. The situation is not always so clear in the case of nonabelian groups.
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} however 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 rather 'inheritable'; since
As a strengthening of solvability, a group G is called supersolvable if it 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 group
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