A study on the structure of finite groups with \(c\)-subnormal subgroups (Q6618054)

Let \(G\) be a finite group. A subgroup \(H \leq G\) is \(c\)-normal in \(G\) if there exists \(T \trianglelefteq G\) such that \(G=HT\) and \(T \cap H \leq H_{G}\), where \(H_{G}\) is the core of \(H\) in \(G\). Similarly, \(H\) is a \(c\)-subnormal subgroup in \(G\) if there exists a subnormal subgroup \(T\) of \(G\) such that \(G=HT\), and \(T \cap H \leq H_{G}\).\N\NThe purpose of the paper under review is to extend some known results on \(c\)-normal subgroups to \(c\)-subnormal subgroups. In particular, the authors prove two theorems that answer the question of what conditions \(c\)-subnormal subgroups must hold so that \(G\) is an element in a formation \(\mathfrak{U}\) of supersoluble groups (see Theorem 3.1 and Theorem 3.2).