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Keywords:
finite group; hypercyclically embedded; subgroup; Sylow subgroup; fusion system
finite group; hypercyclically embedded; subgroup; Sylow subgroup; fusion system
Summary:
Given a prime $p$ and a subgroup $A$ of a finite group $G$, we say that $A$ is a \hbox {$p$-${\rm CAP}$-subgroup} of $G$ if $A$ covers or avoids every $p$-$G$-chief factor, where a $p$-$G$-chief factor is a $G$-chief factor of order divisible by $p$. We say that $A$ is a strong $p$-${\rm CAP}$-subgroup of $G$ if $A$ is a $p$-${\rm CAP}$-subgroup of any subgroup of $G$ containing $A$. We use the concept of strong $p$-${\rm CAP}$-subgroups to investigate the $p\mathfrak {F}$-hypercentrally embedded property of normal subgroups of a finite group and obtain some new results. Moreover, we extend the concept of (strong) $p$-${\rm CAP}$-subgroups to fusion systems and use this to characterize supersolvable and nilpotent fusion systems.
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