On \(T^*\)-groups (Q1307437)

Let \(G\) be a finite group and assume \(H\triangleleft K\triangleleft G\) always implies that \(H\) is \(\pi\)-quasinormal in \(G\). Such a group is called \(T^*\)-group. The authors study finite solvable \(T^*\)-groups. They give several characterizations of such groups. They prove among others that the \(T^*\) property is inherited to all subgroups and give sufficient conditions for \(G\) to be \(T^*\) provided that \(G/H\) is solvable \(T^*\).