(
G
,
H
,
H
0
,
π
)
(G,H,{H_0},\pi )
denotes the following configuration:
H
H
and
H
0
{H_0}
are the subgroups of the finite group
G
G
with
H
0
⊴
H
{H_0} \trianglelefteq H
is the set of primes dividing
(
H
:
H
0
)
(H:{H_0})
. For
(
G
,
H
,
H
0
,
π
)
(G,H,{H_0},\pi )
we consider conditions
(
A
)
({\text {A}})
,
(
B
0
)
({{\text {B}}_0})
, and
(
C
)
({\text {C}})
:
(
A
)
({\text {A}})
Any two
π
\pi
-elements of
H
−
H
0
H - {H_0}
which are
G
G
-conjugate are
H
H
-conjugate.
(
B
0
)
({{\text {B}}_0})
For each
π
\pi
-element
x
∈
H
−
H
0
x \in H - {H_0}
,
C
G
(
x
)
=
I
(
x
)
C
H
(
x
)
{C_G}(x) = I(x){C_H}(x)
where
I
(
x
)
I(x)
is a normal
π
′
\pi ’
-subgroup of
C
G
(
x
)
{C_G}(x)
.
(
C
)
|
(
H
−
H
0
)
G
,
π
|
=
(
G
:
H
)
|
H
−
H
0
|
({\text {C}})\left | {{{(H - {H_0})}^{G,\pi }}} \right | = (G:H)\left | {H - {H_0}} \right |
. We show that if
(
G
,
H
,
H
0
,
π
)
(G,H,{H_0},\pi )
satisfies
(
B
0
)
({{\text {B}}_0})
and
(
C
)
({\text {C}})
, or
(
A
)
({\text {A}})
and
(
B
0
)
({{\text {B}}_0})
, and if
H
/
H
0
H/{H_0}
is solvable, then there is a unique relative normal complement
G
0
{G_0}
of
H
H
over
H
0
{H_0}
.