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 N± 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.