On $S$-quasinormally embedded subgroups of~finite groups
Algebra and discrete mathematics, Tome 13 (2012) no. 1, pp. 18-25
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Let $G$ be a finite group. A subgroup $A$ is called:
1) $S$-quasinormal in $G$ if $A$ is permutable with all Sylow subgroups in $G$
2) $S$-quasinormally embedded in $G$ if every Sylow subgroup of $A$ is a Sylow subgroup of some
$S$-quasinormal subgroup of $G$. Let $B_{seG}$ be the subgroup generated by all the
subgroups of $B$ which are
$S$-quasinormally embedded in $G$.
A subgroup $B$ is called $SE$-supplemented in $G$ if there exists a
subgroup $T$ such that $G=BT$ and
$B\cap T\le B_{seG}$. The main result of the paper is the
following.
Theorem. Let $H$ be a normal subgroup in $G$, and $p$
a prime divisor of $|H|$ such that $(p-1,|H|)=1$.
Let $P$ be a Sylow $p$-subgroup in $H$. Assume
that all maximal subgroups in $P$ are $SE$-supplemented in $G$. Then $H$
is $p$-nilpotent and all its $G$-chief $p$-factors are cyclic.
Keywords:
Finite group, $p$-nilpotent, $S$-quasinormal subgroup.
@article{ADM_2012_13_1_a2,
author = {Kh. A. Al-Sharo and Olga Shemetkova and Xiaolan Yi},
title = {On $S$-quasinormally embedded subgroups of~finite groups},
journal = {Algebra and discrete mathematics},
pages = {18--25},
publisher = {mathdoc},
volume = {13},
number = {1},
year = {2012},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ADM_2012_13_1_a2/}
}
TY - JOUR AU - Kh. A. Al-Sharo AU - Olga Shemetkova AU - Xiaolan Yi TI - On $S$-quasinormally embedded subgroups of~finite groups JO - Algebra and discrete mathematics PY - 2012 SP - 18 EP - 25 VL - 13 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/ADM_2012_13_1_a2/ LA - en ID - ADM_2012_13_1_a2 ER -
Kh. A. Al-Sharo; Olga Shemetkova; Xiaolan Yi. On $S$-quasinormally embedded subgroups of~finite groups. Algebra and discrete mathematics, Tome 13 (2012) no. 1, pp. 18-25. http://geodesic.mathdoc.fr/item/ADM_2012_13_1_a2/