ID 19

1C.8. Let P be a Sylow p-subgroup of G. Show that for every nonnegative

integer a, the numbers of subgroups of order

pa

in P and in G are congruent

modulo p.

Note. If

pa

= |P|, then the number of subgroups of order

pa

in P is clearly

1, and it follows that the number of such subgroups in G is congruent to 1

modulo p. This provides a somewhat different proof that np(G) = 1 mod p.

It is true in general that if

pa

\P\, then the number of subgroups of order

pa

in P is congruent to 1 modulo p, and thus it follows that if

pa

divides the

order of an arbitrary finite group G, then the number of subgroups of order

pa

in G is congruent to 1 mod p.

I D

We now digress from our study of Sylow theory in order to review some

basic facts about p-groups and nilpotent groups. Also, we discuss the Fitting

subgroup, and in the problems at the end of the section, we present some

results about the Frattini subgroup.

Although p-groups are not at all typical of finite groups in general, they

play a prominent role in group theory, and they are ubiquitous in the study

of finite groups. This ubiquity is, of course, a consequence of the Sylow

theorems, and perhaps that justifies our digression.

We should mention that although their structure is atypical when com-

pared with finite groups in general, p-groups are, nevertheless, extremely

abundant in comparison with non-p-groups. There are, for example, 2,328

isomorphism types of groups of order 128 =

27;

the number of types of order

256 =

28

is 56,092; for 512 =

29

the number is 10,494,213; and there are

exactly 49,487,365,422 isomorphism types of groups of order 1,024 =

210.

(These numbers were computed by a remarkable algorithm for counting p-

groups that was developed by E. O'Brien.)

There is an extensive theory of finite p-groups (and also of their infinite

cousins, pro-p-groups), and there are several books entirely devoted to them.

Our brief presentation here will be quite superficial; later, we study p-groups

a bit more deeply, but still, we shall see only a tiny part of what is known.

Perhaps the most fundamental fact about p-groups is that nontrivial

finite p-groups have nontrivial centers. (By our definition, "p-group" means

"finite p-group", but we included the redundant adjective in the previous

sentence and in what follows in order to stress the fact that finiteness is

essential here. Infinite p-groups can have trivial centers, and in fact, they

can be simple groups.)

In fact, a stronger statement is true.