Bulletin of the
Korean Mathematical Society BKMS

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be an $ss$-quasinormal subgroup of $G$ if there is a subgroup $B$ of $G$ such that $G=HB$ and $H$ permutes with every Sylow subgroup of $B$; $H$ is said to be semi-$p$-cover-avoiding in $G$ if there is a chief series $1=G_0 < G_1 < \cdots < G_t=G$ of $G$ such that, for every $i=1, 2,\ldots, t$, if $G_{i}/G_{i-1}$ is a $p$-chief factor, then $H$ either covers or avoids $G_{i}/G_{i-1}$. We give the structure of a finite group $G$ in which some subgroups of $G$ with prime-power order are either semi-$p$-cover-avoiding or $ss$-quasinormal in $G$. Some known results are generalized.

Keywords: $ss$-quasinormal subgroup, semi-$p$-cover-avoiding subgroup, saturated formation

MSC numbers: 20D10, 20D20