Finite Groups with Generalized Subnormal \(\mathfrak{F}\)-Critical Subgroups

Let \(G\) be a finite group and let \(A\) be a subgroup of \(G\) . Let \(A_{\operatorname{sn}G}\) be the subgroup of \(A\) generated by all subnormal subgroups of \(G\) contained in \(A\) , and let \(A^{\operatorname{sn}G}\) be the intersection of all subnormal subgroups of \(G\) containing \(A\) . Let \(N\le G\) . Then we say that \(A\) is \(N\) -subnormal in \(G\) if \(N\) avoids every composition factor \(H/K\) of \(G\) between \(A_{\operatorname{sn}G}\) and \(A^{\operatorname{sn}G}\) , i.e., \(N\cap H= N\cap K\) . In this paper, we give applications of \(N\) -subnormality to the theory of groups with given \(\mathfrak{F}\) -critical subgroups. In particular, using this notion, we give new characterizations of finite solvable groups, metanilpotent groups, and groups with nilpotent derived subgroup \(G'\) .