Frattini's argument explained
In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who used it in a paper from 1885 when defining the Frattini subgroup of a group. The argument was taken by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.[1]
Frattini's argument
Statement
If
is a finite group with normal subgroup
, and if
is a
Sylow p-subgroup of
, then
where
denotes the
normalizer of
in
, and
means the
product of group subsets.
Proof
The group
is a Sylow
-subgroup of
, so every Sylow
-subgroup of
is an
-conjugate of
, that is, it is of the form
for some
(see
Sylow theorems). Let
be any element of
. Since
is normal in
, the subgroup
is contained in
. This means that
is a Sylow
-subgroup of
. Then, by the above, it must be
-conjugate to
: that is, for some
and so
Thus
and therefore
. But
was arbitrary, and so
Applications
- Frattini's argument can be used as part of a proof that any finite nilpotent group is a direct product of its Sylow subgroups.
- By applying Frattini's argument to
, it can be shown that
whenever
is a finite group and
is a Sylow
-subgroup of
.
- More generally, if a subgroup
contains
for some Sylow
-subgroup
of
, then
is self-normalizing, i.e.
.
External links
References
- Book: Hall
, Marshall
. Marshall Hall (mathematician) . The theory of groups . Macmillan . 1959 . New York, N.Y. . (See Chapter 10, especially Section 10.4.)
Notes and References
- M. Brescia, F. de Giovanni, M. Trombetti, "The True Story Behind Frattini’s Argument", Advances in Group Theory and Applications 3, doi:10.4399/97888255036928
