20 1. Sylow Theory
1.19. Theorem . Let P be a finite p-group and let N be a nonidentity normal
subgroup of P. Then N D Z(P) 1. In particular, if P is nontrivial, then
Z(P) 1.
Proof. Since N P , we can let P act on iV by conjugation, and we observe
that N n Z(P) is exactly the set of elements of N that lie in orbits of size
1. By the fundamental counting principle, every orbit has p-power size, and
so each nontrivial orbit (i.e., orbit of size exceeding 1) has size divisible by
p. Since the set N (N n Z(P)) is a union of such orbits, we see that
\N\ -\ND Z(P) | is divisible by p, and thus \N n Z(P) | = \N\ = 0 mod
p, where the second congruence follows because N is a nontrivial subgroup.
Now N H Z(P) contains the identity element, and so |JV D Z(P) | 0. It
follows that \N fl Z(P) | p 1, and hence N n Z(P) is nontrivial, as
required. The final assertion follows by taking N P. M
It is now easy to show that (finite, of course) p-groups are nilpotent,
and thus we can obtain additional information about p-groups by studying
general nilpotent groups. But first, we review some definitions.
A finite collection of normal subgroups Ni of a (not necessarily finite)
group G is a normal series for G provided that
1 = N0 C Ni C C Nr = G.
This normal series is a central series if in addition, we have Ni/Ni-i C
Z(G/Ni-i) for 1 i r. Finally, a group G is nilpotent if it has a central
series. It is worth noting that subgroups and factor groups of nilpotent
groups are themselves nilpotent, although we omit the easy proofs of these
facts.
Given any group G?, we can attempt to construct a central series as fol-
lows. (But of course, this attempt is doomed to failure unless G is nilpotent.)
We start by defining ZQ = 1 and Z\ Z(G). The second center Z2 is
defined to be the unique subgroup such that Z2/Z1 Z{G/Z\). (Note that
Zi exists and is normal in G by the correspondence theorem.) We continue
like this, inductively defining Zn for n 0 so that ZnjZn-\ Z(G/Zn-i).
The chain of normal subgroups
1 = Z0 C Zi C Z2 C
constructed this way is called the upper central series of G. We hasten
to point out, however, that in general, the upper central series may not
actually be a central series for G because it may happen that Z{ G for all
i. In other words, the upper central series may never reach the whole group
G. But if Zr G for some integer r, then { Z ^ | 0 i r } i s a true central
series, and G is nilpotent.
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