|
Finite generalized soluble groups
J. Huanga, B. Hua, A. N. Skibab a School Math. Stat., Jiangsu Normal Univ., Xuzhou,
221116, P. R. CHINA
b Gomel State University named after Francisk Skorina
Abstract:
Let $\sigma =\{\sigma_{i} \mid i\in I\}$ be a partition of the set of
all primes $\mathbb{P}$ and $G$ a finite group. Suppose $\sigma
(G)=\{\sigma_{i} \mid \sigma_{i}\cap \pi (G)\ne \varnothing\}$. A set
$\mathcal{H}$ of subgroups of $G$ is called a complete Hall
$\sigma $-set of $G$ if every nontrivial member of $\mathcal{H}$ is a $\sigma_{i}$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$
contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $i$
such that $\sigma_{i}\in \sigma (G)$. A group $G$ is $\sigma$-full
if $G$ possesses a complete Hall $\sigma $-set. A complete Hall
$\sigma $-set $\mathcal{H}$ of $G$ is called a $\sigma$-basis of $G$ if
every two subgroups $A, B \in\mathcal{H}$ are permutable, i.e., $AB=BA$.
In this paper, we study properties of finite groups having a
$\sigma$-basis. It is proved that if $G$ has a $\sigma$-basis, then
$G$ is generalized $\sigma$-soluble, i.e, $|\sigma (H/K)|\leq 2$ for
every chief factor $H/K$ of $G$. Moreover, every
complete Hall $\sigma$-set of a $\sigma$-full group $G$ forms a
$\sigma$-basis of $G$ iff $G$ is generalized $\sigma$-soluble, and
for the automorphism group $G/C_{G}(H/K)$ induced by $G$ on any its
chief factor $H/K$, we have $|\sigma (G/C_{G}(H/K))|\leq 2$ and also
$\sigma(H/K)\subseteq
\sigma (G/C_{G}(H/K))$ in the case $|\sigma
(G/C_{G}(H/K))|= 2$.
Keywords:
finite group, Hall subgroup, $\sigma$-soluble subgroup,
$\sigma$-basis, generalized ${\sigma}$-soluble group.
Received: 31.01.2018 Revised: 09.07.2019
Citation:
J. Huang, B. Hu, A. N. Skiba, “Finite generalized soluble groups”, Algebra Logika, 58:2 (2019), 252–270; Algebra and Logic, 58:2 (2019), 173–185
Linking options:
https://www.mathnet.ru/eng/al893 https://www.mathnet.ru/eng/al/v58/i2/p252
|
|