Geodesic
Criteria for the $p$-Solvability and~$p$-Supersolvability of Finite Groups
Matematičeskie zametki, Tome 94 (2013) no. 3, pp. 455-472

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Let $A$, $K$, and $H$ be subgroups of a group $G$ and $K\leqslant H$. Then we say that $A$ covers the pair $(K,H)$ if $AH=AK$ and avoids the pair $(K,H)$ if $A\cap H=A\cap K$. A pair $(K,H)$ in $G$ is said to be maximal if $K$ is a maximal subgroup of $H$. In the present paper, we study finite groups in which some subgroups cover or avoid distinguished systems of maximal pairs of these groups. In particular, generalizations of a series of known results on (partial) $CAP$-subgroups are obtained.
Mots-clés : solvable group, supersolvable group, maximal pair
Keywords: weakly $CAP_{p}$-subgroup, weakly $CAP$-subgroup, (conditional) cover-avoiding property of subgroups, (partial) $CAP$-subgroup. 
@article{MZM_2013_94_3_a11,
     author = {Yufeng Liu and Guo Wenbin and V. A. Kovaleva and A. N. Skiba},
     title = {Criteria for the $p${-Solvability} and~$p${-Supersolvability} of {Finite} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {455--472},
     publisher = {mathdoc},
     volume = {94},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_94_3_a11/}
}
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Yufeng Liu; Guo Wenbin; V. A. Kovaleva; A. N. Skiba. Criteria for the $p$-Solvability and~$p$-Supersolvability of Finite Groups. Matematičeskie zametki, Tome 94 (2013) no. 3, pp. 455-472. http://geodesic.mathdoc.fr/item/MZM_2013_94_3_a11/