# How to find a normal abelian subgroup in a solvable group?

Possible Duplicate:
A Nontrivial Subgroup of a Solvable Group

If $H$ is nontrivial normal subgroup of the solvable group $G$, then how can I show that there is a nontrivial subgroup $A\leq H$ such that $A$ is abelian and normal in $G$?

I am looking for hints so that I can create my own solution.

Thank you all.

#### Solutions Collecting From Web of "How to find a normal abelian subgroup in a solvable group?"

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 $\,H\,$ …!)