Geodesic
On some generalizations of permutability and $S$-permutability
Problemy fiziki, matematiki i tehniki, no. 4 (2013), pp. 47-54

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Let $H$ and $X$ be subgroups of a finite group $G$. Then we say that $H$ is: $X$-quasipermutable (respectively, $X_S$-quasipermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) $V$ of $B$ such that $(|H|,|V|)=1$; $X$-propermutable (respectively, $X_S$-propermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) of $B$. In this paper we analyze the influence of $X$-quasipermutable, $X_S$-quasipermutable, $X$-propermutable and $X_S$-propermutable subgroups on the structure of $G$.
Keywords: finite group, $X$-quasipermutable subgroup, $X_S$-quasipermutable subgroup, $X$-propermutable subgroup, $X_S$-propermutable subgroup, Sylow subgroup, Hall subgroup, $p$-supersoluble group, maximal subgroup, saturated formation, $PST$-group
Mots-clés : $p$-soluble group, $PT$-group. 
@article{PFMT_2013_4_a7,
     author = {Xiaolan Yi and A. N. Skiba},
     title = {On some generalizations of permutability and $S$-permutability},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {47--54},
     publisher = {mathdoc},
     number = {4},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2013_4_a7/}
}
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Xiaolan Yi; A. N. Skiba. On some generalizations of permutability and $S$-permutability. Problemy fiziki, matematiki i tehniki, no. 4 (2013), pp. 47-54. http://geodesic.mathdoc.fr/item/PFMT_2013_4_a7/