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.
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