I refer to Martin Isaac book, "Finite Group Theory", Theorem 2.12 pag. 55. In the proof there is a detail I can't understand.
Our hypotheses are the following: $G$ finite group, $H\leq G$ s.t. $\langle H,H^x\rangle$ is nilpotent $\forall x\in G$.
We want to show that $H\lhd\lhd G$ by induction on $|G|$. The case $|G|=1$ is trivial. Then we suppose the statement true for every proper subgroup of $G$, i.e. if $H\leq K\lneq G$ then $H\lhd\lhd K$. And till here everything is all right. The detail I can't understand, is why the author (and my teacher too) says: if $H\leq K\lneq G$, then $\langle H,H^x\rangle$ is nilpotent $\forall x\in K$, HENCE $H\lhd\lhd K$... like the fact that $\langle H,H^x\rangle$ is nilpotent $\forall x\in K$ was important to conclude that $H\lhd\lhd K$.
I can't understand what is the role played by the nilpotency of $\langle H,H^x\rangle$ in saying that $H\lhd\lhd K$; wasn't this true simply by inductive hypothesis? Thank you all
I think you are misunderstanding the induction hypothesis. You are not assuming that $H$ is subnormal in all the proper subgroups of $G$. The induction hypothesis is that, for all groups (say $U$) of order smaller than the order of $G$, whenever a subgroup $V$ is such that $\langle V, V^{x}\rangle$ is nilpotent for all $x\in U$, then $V$ is subnormal in $U$. Now apply this with $U = K$ in the notation of the proof. Since $\langle H,H^{x}\rangle$ is nilpotent for all $x\in G$, it is, a fortiori, true that $\langle H,H^{x}\rangle$ is nilpotent for all $x\in K$. Thus, by the induction hypothesis (with $U=K$, where $K$ is a proper subgroup of $G$, and with $V=H$), you can conclude that $H$ is subnormal in $K$. (Then, continue with the zipper lemma, etc., as in the text.)
I'm not really saying anything new here, just emphasising a few things. I hope it helps.