22 1. Sylow Theory

Proof. We prove that Ni C Z{ by induction on i. Since Zo — 1 = No, we

can suppose that i 0, and by the inductive hypothesis, we can assume

that Ni-i C Zi_i. For notational simplicity, write T V = A^_i and Z — Z{-\,

and observe that since N C Z, there is a natural surjective homomorphism

0 : G/N - G/Z, defined by 6(Ng) = Zg for elements g G G. (This map

is well defined since Zg is the unique coset of Z that contains Ng, and so

0(Ng) depends only on the coset Ng and not on the element g.)

Since 0 is surjective, it carries central elements of G/N to central ele-

ments of G/Z, and since Ni/N is central in G/N, it is mapped by 0 into

Z(G/Z) = Zi/Z. If x G Ni, therefore, it follows that Zx = 0(Nx) G Z{/Z,

and thus x £ Zi, &s required. •

If G is an arbitrary nilpotent group, then G is a term of its upper central

series, which, therefore is a true central series. We have G = Zr for some

integer r 0, and the smallest integer r for which this happens is called the

nilpotence class of G. Thus nontrivial abelian groups have nilpotence class

1, and the groups of nilpotence class 2 are exactly the nonabelian groups G

such that G/Z(G) is abelian.

Since the upper central series of the nilpotent group G is really a central

series, we see that if G has nilpotence class r, then it has a central series of

length r. (In other words, the series has r containments and r + 1 terms.) By

Theorem 1.21, we see that G cannot have a central series of length smaller

than r, and thus the nilpotence class of a nilpotent group is exactly the

length of its shortest possible central series. In some sense, the nilpotence

class can be viewed as a measure of how far from being abelian a nilpotent

group is.

One of the most useful facts about nilpotent groups, and thus also about

finite p-groups, is that "normalizers grow".

1.22. Theorem. Let H G, where G is a (not necessarily finite) nilpotent

group. Then NG(H) H.

Before we proceed with the (not very difficult) proof of Theorem 1.22,

we digress to discuss the "bar convention", which provides a handy notation

for dealing with factor groups. Suppose that N G, where G is an arbitrary

group. We write G to denote the factor group G/N, and we think of the

overbar as the name of the canonical homomorphism from G onto G. If

g G G is any element, therefore, we write ~g to denote the image of g in

G, and so we see that ~g is simply another name for the coset Ng. Also, if

H C G is any subgroup, then H is the image of H in G.

By the correspondence theorem, the homomorphism "overbar" defines a

bijection from the set of those subgroups of G that contain N onto the set

of all subgroups of G. Every subgroup of G, therefore, has the form H for