I try to solve the following problem.
Let $X,Y$ be subgroups of finite group $G$, and $Y$ centralizes $[X,Y]$.
- Prove that $[Y,Y]$ centralizes $X$.
- Now assume that $X$ is normal in $G$. Show that $[X,Y]$ is abelian.
The first item can be solved by applying the three subgroup lemma. But I don't know how to solve the second item. I give here my attempt (not rigorous and maybe not correct) $$ [[X,Y],[X,Y]] = [[X,Y],[Y,X]]\subseteq [X,Y,Y,X] = [[[X,Y],Y],X] =1. $$ We know that for every finite group $F$ the following inclusion is valid $$[F^i,F^j]\subseteq F^{i+j},$$ where $F^i=[F,F,\ldots,F]$ ($i$-times). But I don't know is it possible to prove the analogous inclusion $[[A,B],[C,D]]\subseteq [A,B,C,D]$ for arbitrary subgroups of $F$ or maybe $[[A,B],[A,C]]\subseteq [A,B,A,C]$, when $A$ is normal in $F$.