ID 21
Conversely, if G is nilpotent, the upper central series of G really is a
central series. For finite groups G, this is especially easy to prove.
1.20. Lemma. Let G be finite. Then the following are equivalent.
(1) G is nilpotent.
(2) Every nontrivial homomorphic image of G has a nontrivial center.
(3) G appears as a member of its upper central series.
Proof. We have already remarked that homomorphic images of nilpotent
groups are nilpotent. Also, since the first nontrivial term of a central series
for a nilpotent group is contained in the center of the group, it follows that
nontrivial nilpotent groups have nontrivial centers. This shows that (1)
implies (2).
Assuming (2) now, it follows that if Zi G, where Zi is a term in the
upper central series for G, then Zi+i/Zi = Z(G/Zi) is nontrivial, and thus
Zi Zi+i. Since G is finite and the proper terms of the upper central series
are strictly increasing, we see that not every term can be proper, and this
establishes (3).
Finally, (3) guarantees that the upper central series for G is actually a
central series, and thus G is nilpotent, proving (1).
If P is a finite p-group, then of course, every homomorphic image of P is
also a finite
p-group, and thus every nontrivial homomorphic image of P has
a nontrivial center. It follows by Lemma 1.20, therefore, that finite p-groups
are nilpotent. In fact, we shall see in Theorem 1.26 that much more is true:
a finite group G is nilpotent if and only if every Sylow subgroup of G is
normal.
Next, we show that the terms of the upper central series of a nilpotent
group contain the corresponding terms of an arbitrary central series, and
this explains why the upper central series is called "upper". It also provides
an alternative proof of the implication (1) = (3) of Lemma 1.20, without
the assumption that G is finite.
1.21. Theorem. Let G be a (not necessarily finite) nilpotent group with
central series
1 = N0 C C C Nr = G,
and as usual, let
i = z0cz1cz2c--.
be the upper central series for G. Then Ni C Zi for 0 i r, and in
particular, Zr = G.
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