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*_{n-1}, *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 Jordan-Hölder 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 Jordan-Hölder 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 *n*th 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 non-abelian groups.

A small example of a solvable, non-abelian group is the symmetric group *S*_{3}.
In fact, as the smallest simple non-abelian 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, non-abelian 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

- 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.

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