Let $\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a $\sigma_i$-group for some $i=i(H/K)$. We prove the following Theorem. (i) If $G$ is $\pi$-separable, $H$ is a nilpotent Hall $\pi$-subgroup and $E$ a $\pi$-complement of $G$ such that $EX=XE$ for some subgroup $X$ of $H$ such that $H'\leqslant X\leqslant \Phi(H)$, then $l_\pi(G)\leqslant1$. (ii) If $G$ is $\sigma$-soluble and $\{H_1,\dots, H_t\}$ is a Wielandt $\sigma$-basis of $G$ such that $H_i$ permutes with $H_j$ for all $i$, $j$, then $l_{\sigma_i}(G)\leqslant 1$ for all $i$. (iii) If $G$ is $\sigma$-soluble and $\{H_1,\dots, H_t\}$ is a Wielandt $\sigma$-basis of $G$ such that $H_i$ permutes with $\Phi(H_j)$ for all $i$, $j$, then $l_{\sigma_i}(G)\leqslant 1$ for all $i$. (iv) If $l_\pi(G)\leqslant 1$, then $QX=XQ$ each characteristic subgroup $X$ of $H$ and any Sylow subgroup $Q$ of $G$ such that $HQ=QH$. (v) If $G$ is $\sigma$-soluble with $l_{\sigma_i}\leqslant 1$ for all $i$ and $\{H_1,\dots, H_t\}$ is a $\sigma$-basis of $G$, then each characteristic subgroup of $H_i$ permutes with each characteristic subgroup of $H_j$.
@article{PFMT_2020_4_a16,
author = {N. S. Kosenok and V. M. Selkin},
title = {On the $\sigma_i$-length of a finite $\sigma$-soluble group},
journal = {Problemy fiziki, matematiki i tehniki},
pages = {95--97},
year = {2020},
number = {4},
language = {en},
url = {http://geodesic.mathdoc.fr/item/PFMT_2020_4_a16/}
}
TY - JOUR
AU - N. S. Kosenok
AU - V. M. Selkin
TI - On the $\sigma_i$-length of a finite $\sigma$-soluble group
JO - Problemy fiziki, matematiki i tehniki
PY - 2020
SP - 95
EP - 97
IS - 4
UR - http://geodesic.mathdoc.fr/item/PFMT_2020_4_a16/
LA - en
ID - PFMT_2020_4_a16
ER -
%0 Journal Article
%A N. S. Kosenok
%A V. M. Selkin
%T On the $\sigma_i$-length of a finite $\sigma$-soluble group
%J Problemy fiziki, matematiki i tehniki
%D 2020
%P 95-97
%N 4
%U http://geodesic.mathdoc.fr/item/PFMT_2020_4_a16/
%G en
%F PFMT_2020_4_a16
N. S. Kosenok; V. M. Selkin. On the $\sigma_i$-length of a finite $\sigma$-soluble group. Problemy fiziki, matematiki i tehniki, no. 4 (2020), pp. 95-97. http://geodesic.mathdoc.fr/item/PFMT_2020_4_a16/
