1C 15
A frequently used application of the Sylow C-theorem is the so-called
"Frattini argument", which we are about to present. Perhaps the reason
that this result is generally referred to as an "argument" rather than as a
"lemma" or "theorem" is that variations on its proof are used nearly as often
as its statement.
1.13. Lemma (Frattini Argument). Let N G where N is finite, and sup-
pose that P G Sylp(iV). Then G = NG(P)N.
Proof. Let g G G, and note that
P9
C
N9
= N, and thus
P9
is a subgroup
of N having the same order as the Sylow p-subgroup P. It follows that
P9
G Sylp(iV), and so by the Sylow C-theorem applied in N, we deduce that
(P9)n
= P, for some element ne N. Since
Pgn
= P, we have gn G NG(P),
and so g G
NQ(P)U~1
C
NG(P)N.
But g G G was arbitrary, and we deduce
that G = NG(P)N, as required.
By definition, a Sylow p-subgroup of a finite group G is a p-subgroup
that has the largest possible order consistent with Lagrange's theorem. By
the Sylow E-theorem, we can make a stronger statement: a subgroup whose
order is maximal among the orders of all ^-subgroups of G is a Sylow p-
subgroup. An even stronger assertion of this type is that every maximal p-
subgroup of G is a Sylow p-subgroup. Here, "maximal" is to be interpreted
in the sense of containment: a subgroup H of G is maximal with some
property if there is no subgroup K H that has the property. The truth of
this assertion is the essential content of the Sylow "development" theorem.
1.14. Theorem (Sylow D). Let P be a p-subgroup of a finite group G. Then
P is contained in some Sylow p-subgroup of G.
Proof. Let 5 G Sylp(G). Then by Theorem 1.11, we know that
PCS9
for
some element g G G. Also, since
\S9\
= \S\, we know that
S9
is a Sylow
p-subgroup of G.
Given a finite group G, we consider next the question of how many Sylow
p-subgroups G has. To facilitate this discussion, we introduce the (not quite
standard) notation np(G) = |Sylp(G)|. (Occasionally, when the group we
are considering is clear from the context, we will simply write np instead of
np(G).)
First, by the Sylow C-theorem, we know that Sylp(G) is a single orbit
under the conjugation action of G. The following is then an immediate
consequence.
1.15. Corollary. Let S G Sylp(G), where G is a finite group. Then np(G)
|G:N
G
(5)| .
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