Derived series
The derived series is a particular sequence of decreasing subgroups of a group .
Specifically, let be a group. The derived series is a sequence
defined recursively as
,
, where
is the derived group (i.e., the commutator subgroup) of a group
.
A group for which
is trivial for sufficiently large
is called solvable. The least
such that
is called the solvability class of
. By transfinite recursion, this notion can be extended to infinite ordinals, as well.
Let be the
th term of the lower central series of
. Then from the relation
and induction, we have
In particular, if
is nilpotent of class at most
, then it is solvable of class at most
. Thus if
is nilpotent, then it is solvable; however, the converse is not generally true.
By induction on it follows that if
and
are groups and
is a homomorphism, then
; in particular, if
is surjective,
. It follows that for all nonnegative integers
,
is a characteristic subgroup of
.
If is a decreasing sequence of subgroups such that
is a normal subgroup of
and
is abelian for all integers
, then
, by induction on
.