A group is solvable if there exists some nonnegative integer for which , where is the th term of the derived series of . The least integer satisfying this condition is called the solvability class of . A group is abelian if and only if its solvability class is at most one; it is trivial if and only if its solvability class is zero.
Every nilpotent group is solvable. In particular, if a group is nilpotent of class at most , then it is solvable of class at most .
However, the converse is not true in general. For instance, is solvable of class 2: the first three terms of its derived series are But it is not nilpotent: the terms of its lower central series are In fact, is not even residually nilpotent, i.e., the infinite extension of the lower central series of never reduces to .
In 1962, Walter Feit and John Thompson proved that every finite group of odd order is solvable (see Feit-Thompson Theorem). This result arose from a conjecture of William Burnside, and earlier work by Michio Suzuki.
Characteristics of Solvable Groups
Proposition. Let be a group, and let be a positive integer. Then the following four conditions are equivalent.
- is solvable of class at most ;
- There exists a decreasing sequence of normal subgroups of such that , , and is abelian for every index ;
- There exists a decreasing sequence of subgroups of such that , , normalizes , and the quotient group is abelian for every index ;
- There exists an abelian normal subgroup of for which is solvable of class at most .
Proof. To show that (1) implies (2), we may take . Also, (2) evidently implies (3). To show that (3) implies (1), we note by induction that , for each index ; hence .
To show that (1) implies (4), we may take . To show that (4) implies (1), we define to be the canonical homomorphism from to . Then ; since is commutative, . This completes the proof.
Thus a group is solvable if and only if it can be obtained by iterative extension by abelian groups.
Proof. A finite simple group is abelian if and only if it is cyclic and of prime order. Thus if the quotient groups of a Jordan-Hölder series of a group are cyclic and of prime order, then satisfies condition (3) of the proposition and hence is solvable.
Conversely, if is solvable, then it has a composition series whose quotients are abelian. Hence the quotients of the Jordan-Hölder series derived from this composition series are abelian, so they are cyclic and of prime order.