Solvable group
A solvable group is a type of group of particular interest, particularly in Galois theory.
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.
Corollary. A finite group is solvable if and only if every quotient of its Jordan-Hölder series is a cyclic group of prime order.
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.