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 $$G'$$ .