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)| .