My question is about the induction method. This was in a theorem that I read. Let H be a normal subgroup of a finite group G. If G satisfies H-some statement then G is solvable. In the poof I have this. Let G be a counter example of minimal order. Let N be a proper normal subgroup of G. Since N satisfies the H-some statement then N is solvable by the induction in the order of G. Here is my question. H may not be a subgroup of N so, how did he apply the induction method?