Since
L
∞
(
S
n
) =
A
n
and
L
∞
(
A
n
) =
A
n
,
E
i
=
A
n
for
i
≥
1. There is no Fitting length.
Theorem 3.8.
A finite group has a Fitting length if and only if it is solvable.
Proof.
Suppose
G
is a finite solvable group.
If
F
i
6
=
G
then
G/F
i
is a nontrivial finite
solvable group so its Fitting subgroup
F
i
+1
/F
i
is nontrivial. Therefore
F
i
+1
6
=
F
i
, so for
large
i
we must have
F
i
=
G
.
Similarly, if
E
i
6
=
{
e
}
then
E
i
+1
6
=
E
i
(because
E
i
+1
=

SUBGROUP SERIES II
7
L
∞
(
E
i
)
⊂
L
1
(
E
i
) =
E
0
i
, which is a proper subgroup of
E
i
because
E
i
is a nontrivial solvable
group). Therefore
E
i
is trivial for large
i
.
Now suppose
G
has a Fitting length. The upper nilpotent series is a normal series for
G
with nilpotent factors. Since nilpotent groups are solvable and solvability of
N
and
H/N
implies solvability of
H
, we get solvability of
G
by arguing inductively that every
F
i
is
solvable.
Remark 3.9.
A group
G
is called
metacyclic
,
metabelian
, or
metanilpotent
if it has a normal
subgroup
N
such that
N
and
G/N
are both cyclic, both abelian, or both nilpotent. This
means the normal series
{
e
}
C
N
C
G
has both factors cyclic or abelian or nilpotent, so
these properties are preserved by passage to subgroups and quotient groups (but not direct
products, since the direct product need not have such a series with 2 factors).
All such
groups are solvable since
N
and
G/N
are solvable. Check that metabelian is the same as
having solvable length
≤
2 while (when
G
is finite) metanilpotent is the same as having
Fitting length
≤
2.
In addition to the Fitting subgroup, there is another important nilpotent subgroup of
any finite group: the Frattini subgroup.
Definition 3.10.
The
Frattini
subgroup of a finite group is the intersection of its maximal
subgroups:
Φ(
G
) =
\
max.
M
M.
We set the trivial group to have trivial Frattini subgroup.
The intersection defining Φ(
G
) is preserved by conjugations, so Φ(
G
)
C
G
. For instance,
Φ(
D
4
) =
{
1
, r
2
}
and Φ(
S
n
) is trivial for
n
≥
3.
(In particular, since
D
4
is the 2-Sylow
subgroup of
S
4
, we see that if
H
⊂
G
then Φ(
H
) can be larger than Φ(
G
).) When
G
is
nilpotent,
G
0
⊂
Φ(
G
) by Theorem 2.2(2), so the quotient
G/
Φ(
G
) is abelian. Much more
can be said (and used) about this quotient when
G
is a finite
p
-group: look up the “Burnside
basis theorem” in a group theory book.
Theorem 3.11
(Frattini, 1885)
.
For any finite group
G
,
Φ(
G
)
is nilpotent.
Proof.
We will show all the Sylow subgroups of Φ(
G
) are normal subgroups of Φ(
G
). Let
P
be a Sylow subgroup of Φ(
G
). Then
G
= Φ(
G
) N
G
(
P
) (Frattini argument; see the handout
on applications of the Sylow theorems). If
P
is not normal in
G
then N
G
(
P
)
6
=
G
. Let
M
be
a maximal subgroup of
G
containing N
G
(
P
), so Φ(
G
)
⊂
M
. Therefore
G
= Φ(
G
) N
G
(
P
)
⊂
M
, a contradiction, so
P
C
G
, which implies
P
C
Φ(
G
).
Since, for any finite group
G
, Φ(
G
) is a nilpotent normal subgroup of
G
,
Φ(
G
)
⊂
F(
G
)
.

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- Group Theory, Normal subgroup, Subgroup, KEITH CONRAD, Prime Index