A note on $c$-normal subgroups of finite groups

Abstract

Let $G$ be a finite group. We fix in  every non-cyclic Sylow subgroup  $P$ of $G$ some its subgroup $D$ satisfying $1<|D|<|P|$  and study the structure of $G$ under assumption that all subgroups $H$  of  $P$  with  $|H|=|D|$  are  $c$-normal in $G$.