Abstract
Let 𝐺 be a finite group, and let 𝔉 be a hereditary saturated formation.
We denote by
Z
F
(
G
)
\mathbf{Z}_{\mathfrak{F}}(G)
the product of all normal subgroups 𝑁 of 𝐺 such that every chief factor
H
/
K
H/K
of 𝐺 below 𝑁 is 𝔉-central in 𝐺, that is,
(
H
/
K
)
⋊
(
G
/
C
G
(
H
/
K
)
)
∈
F
(H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F}
.
A subgroup
A
⩽
G
A\leqslant G
is said to be 𝔉-subnormal in the sense of Kegel, or 𝐾-𝔉-subnormal in 𝐺, if there is a subgroup chain
A
=
A
0
⩽
A
1
⩽
⋯
⩽
A
n
=
G
A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=G
such that either
A
i
-
1
⊴
A
i
A_{i-1}\trianglelefteq A_{i}
or
A
i
/
(
A
i
-
1
)
A
i
∈
F
A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F}
for all
i
=
1
,
…
,
n
i=1,\ldots,n
.
In this paper, we prove the following generalization of Schenkman’s theorem on the centraliser of the nilpotent residual of a subnormal subgroup:
Let 𝔉 be a hereditary saturated formation containing all nilpotent groups, and let 𝑆 be a 𝐾-𝔉-subnormal subgroup of 𝐺.
If
Z
F
(
E
)
=
1
\mathbf{Z}_{\mathfrak{F}}(E)=1
for every subgroup 𝐸 of 𝐺 such that
S
⩽
E
S\leqslant E
, then
C
G
(
D
)
⩽
D
\mathbf{C}_{G}(D)\leqslant D
, where
D
=
S
F
D=S^{\mathfrak{F}}
is the 𝔉-residual of 𝑆.
Finite groups with given systems of generalised σ-permutable subgroups