TITLE: A Nontrivial Subgroup of a Solvable Group
QUESTION [15 upvotes]: Question: Let $G$ be a solvable group, and let $H$ be a nontrivial normal subgroup of $G$. Prove that there exists a nontrivial subgroup $A$ of $H$ that is Abelian and normal in $G$.
[ref: this is exercise 11 on page 106 of [DF] := Dummit and Foote's Abstract Algebra, 3rd edition]
UPDATE: I cannot use the derived series of a group (the concept is not introduced in [DF] until some 90 pages later), and the use of a commutator subgroup is questionable.

REPLY [3 votes]: Hints (for you to prove):
1) It is true that
$$H\geq H'\geq\ldots\geq H^{(n)}=1\,\, ,\,\, \text{for some}\,\,\,n\in\Bbb N$$
2) Show that $\,H^{(n-1)}\triangleleft G\,\;\;$ (Yes, not only in $\,H\,$ ...!)