As an example, the derived series is a normal series for any group . If is nilpotent, the ascending central series is a normal series for .

A commonly used fact for composition series: When , is simple iff is maximal in the set of all normal subgroups of with .

*Proof.* (i) Let
be a maximal normal subgroup of
. Then
is simple. Let
be a maximal normal subgroup of
and so on. Since
is finite, this process must end with
. Thus
is a composition series.

(ii) If is abelian, , then is abelian since it is a subgroup of and is abelian since it is isomorphic to .

(iii) If , is a proper normal subgroup of . All proper normal subgroups of have the form for some . ◻

*Proof.* If
is solvable,
is a solvable series. If
is a solvable series for
, then
abelian implies that
.
abelian implies
. Proceeding by induction conclude
. In particular,
so
is solvable. ◻

*Proof.* A composition series with cyclic factors is a
solvable series. Conversely, assume
is a solvable series for
. If
, let
be a maximal normal subgroup of
containing
. If
, let
be a maximal normal subgroup of
containing
. Continue until we obtain a series
with each subgroup a maximal normal subgroup of the
preceding, whence each factor is simple. This series terminates as in,
eventually
since
is finite. Doing this for each pair
gives a solvable refinement
of the original series. Each factor of this series is
abelian and simple hence cyclic of prime order. ◻

*Proof.* Since
is a composition series,
has no proper refinements. Thus any refinement of
is obtained by inserting additional copies of
. Any refinement of
has the exact same nontrivial factors as
and is thus equivalent to
. ◻

*Proof.* Note
. Similarly,
. Thus
. Also,
. We will define an epimorphism
with kernel
. This would imply that
and that
. Define
as follows: If
, let
.
implies
. Thus
is well defined.
is clearly surjective.
Finally,
. Hence
.
. A symmetric argument shows (ii) and
whence (iii) follows. ◻

*Proof.* Let
and
be subnormal [resp. normal] series. Let
and for each
, consider the groups
For each
, the Zassenhaus lemma applied to
shows that
(if the original series is normal, each
) Inserting these groups between each
and
, denoting