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
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