On weakly $N$-subnormal subgroups of finite groups

Let G be a finite group and A, N\leq G . Then A_{\operatorname{sn} G} is the subnormal core of A in G , that is, the subgroup of A generated by all subnormal subgroups of G , contained in A ; A^{\operatorname{sn} G} is the subnormal closure of A in G , that is, the intersection of all subnormal subgroups of G containing A . We say that a subgroup A of G is (i) N -subnormal in G if N\cap A^{\operatorname{sn} G}=N\cap A_{\operatorname{sn} G} ; (ii) weakly N -subnormal in G if for some subnormal subgroup T of G we have AT=G and A\cap T\leq S\leq A , where S is N -subnormal in G . In this paper, we consider some applications of these two concepts. In particular, we prove that a finite group G is soluble if and only if G has a normal subgroup N with soluble factor G/N such that in each maximal chain M_{3}< M_{2}< M_{1}< M_{0}= G of G of length 3 , at least one of the subgroups M_{3} , M_{2} or M_{1} is weakly N -subnormal in G .