Finite soluble groups in which all \(n\)-maximal subgroups
are \(\cal F\)-subnormal
V. Kovalyova, A. Skiba
Throughout this paper, all groups are finite.
We use \(\cal U\), \(\cal N\), and \({\cal N}^{r}\)
to denote the class of all supersoluble groups, the class of all nilpotent
groups, and the class of soluble groups of nilpotent length at most
\(r\) (\(r\geq 1\)).
Recall that a subgroup \(H\) of \(G\) is called a \(2\)-maximal
(second maximal) subgroup of \(G\) whenever \(H\) is
a maximal subgroup of some maximal subgroup \(M\) of \(G\).
Similarly, we can define \(3\)-maximal subgroups,
and so on.
Let \(\cal F\) be a non-empty formation.
Recall that a subgroup \(H\) of a group \(G\) is said to be
\(\cal F\)-subnormal in \(G\) if either \(H=G\) or there exists a chain of subgroups
\[H=H_0 < H_1 < \ldots < H_{n}=G\]
such that \(H_{i-1}\) is a maximal subgroup of
\(H_i\) and \(H_{i}/(H_{i-1})_{H_i}\in \cal F\), for \(i=1,\ldots,n\).
We study the groups in which all \(n\)-maximal subgroups
are \(\cal F\)-subnormal. The following theorems are proved.
Theorem A.Let \(\cal F\) be an \(r\)-multiply saturated formation
such that \({\cal N}\subseteq {\cal F}\subseteq {{\cal N}}^{r + 1}\) for some \(r \geq 0\).
If every \(n\)-maximal subgroup of a soluble group \(G\)
is \(\cal F\)-subnormal in \(G\) and \(|\pi (G)|\geq n+r+1\), then \(G\in \cal F\).
Theorem B.Let \({\cal F}=LF(F)\) be a saturated formation such that
\({\cal N}\subseteq {\cal F}\subseteq \cal U\), where
\(F\) is the canonical local satellite of \(\cal F\).
Let \(G\) be a soluble group with \(|\pi (G)|\geq n+1\).
Then all \(n\)-maximal subgroups of \(G\)
are \(\cal F\)-subnormal in \(G\) if and only if \(G\) is a group of one of
the following types:
I. \(G\in \cal F\).
II. \(G=A\rtimes B\), where \(A=G^{\cal F}\) and \(B\) are Hall subgroups of \(G\),
while \(G\) is Ore dispersive and satisfies the following:
(1) \(A\) is either of the form \(N_1\times \ldots \times N_t\),
where each \(N_i\) is a minimal normal subgroup of \(G\), which is a Sylow
subgroup of \(G\), for \(i=1, \ldots , t\), or a Sylow \(p\)-subgroup of \(G\) of exponent \(p\)
for some prime \(p\) and the commutator subgroup, the Frattini subgroup, and
the center of \(A\) coincide, while \(A/\Phi (A)\) is an \(\cal F\)-eccentric chief factor
of \(G\);
(2) every
\(n\)-maximal subgroup of \(G\)
belongs to \(\cal F\) and induces on the Sylow \(p\)-subgroup of
\(A\) an automorphism group which is contained in \(F(p)\)
for every prime divisor \(p\) of \(|A|\).
Theorem C.Let \(\cal F\) be a saturated formation such that
\({\cal N}\subseteq {\cal F}\subseteq \cal U\).
If every \(n\)-maximal subgroup of a soluble group \(G\) is
\(\cal F\)-subnormal in \(G\) and \(|\pi (G)|\geq n\), then \(G\) is
\(\phi\)-dispersive for some ordering \(\phi\) of the set of all primes.
