BRIEF MESSAGE
On the $\mathfrak{F}$-hypercentral subgroups with the sylow tower property of finite groups
V. I. Murashka Francisk Skorina Gomel State University
(Gomel, Republic of Belarus)
Abstract:
Throughout this paper all groups are finite. Let
$A$ be a group of automorphisms of a group
$G$ that contains all inner automorphisms of
$G$ and
$F$ be the canonical local definition of a saturated formation
$\mathfrak{F}$. An
$A$-composition factor
$H/K$ of
$G$ is called
$A$-
$\mathfrak{F}$-central if
$A/C_A(H/K)\in F(p)$ for all
$p\in\pi(H/K)$. The
$A$-
$\mathfrak{F}$-hypercenter of
$G$ is the largest
$A$-admissible subgroup of
$G$ such that all its
$A$-composition factors are
$A$-
$\mathfrak{F}$-central. Denoted by
$\mathrm{Z}_\mathfrak{F}(G, A)$.
Recall that a group
$G$ satisfies the Sylow tower property if
$G$ has a normal Hall
$\{p_1,\dots, p_i\}$-subgroup for all
$1\leq i\leq n$ where
$p_1>\dots>p_n$ are all prime divisors of
$|G|$.
The main result of this paper is: Let
$\mathfrak{F}$ be a hereditary saturated formation,
$F$ be its canonical local definition and
$N$ be an
$A$-admissible subgroup of a group
$G$ where
$\mathrm{Inn}\,G\leq A\leq \mathrm{Aut}\,G$ that satisfies the Sylow tower property. Then
$N\leq\mathrm{Z}_\mathfrak{F}(G, A)$ if and only if
$N_A(P)/C_A(P)\in F(p)$ for all Sylow
$p$-subgroups
$P$ of
$N$ and every prime divisor
$p$ of
$|N|$.
As corollaries we obtained well known results of R. Baer about normal subgroups in the supersoluble hypercenter and elements in the hypercenter.
Let
$G$ be a group. Recall that $L_n(G)=\{ x\in G\,\,| \,\,[x, \alpha_1,\dots, \alpha_n]=1 \,\,\forall \alpha_1,\dots, \alpha_n\in\mathrm{Aut}\,G\}$ and
$G$ is called autonilpotent if
$G=L_n(G)$ for some natural
$n$. The criteria of autonilpotency of a group also follow from the main result. In particular, a group
$G$ is autonilpotent if and only if it is the direct product of its Sylow subgroups and the automorphism group of a Sylow
$p$-subgroup of
$G$ is a
$p$-group for all prime divisors
$p$ of
$|G|$. Examples of odd order autonilpotent groups were given.
Keywords:
Finite group, nilpotent group, supersoluble group, autonilpotent group, $A$-$\mathfrak{F}$-hypercenter of a group, hereditary saturated formation.
UDC:
512.542 Received: 15.06.2018
Accepted: 12.07.2019
Language: English
DOI:
10.22405/2226-8383-2018-20-2-391-398