Proof with solvable groups

Question

I need help with this proof please:

prove: if $G$ is a solvable group then $G$ cross $G$ is a solvable group.

Answer 1

The following proposition is probably in your text:

A group $G$ is solvable iff $G^{(n)} = 1$ for some $n$. (Here $G^{n}$ denotes the $nth$ derived subgroup, $G^{(0)} = G$, $G^{(1)} = [G,G]$ ... $G^{(n+1)} = (G^{(n)})^{(1)}$.)

You should be able to show that $ G^{(i)} \times H^{(i)} = (G \times H)^{(i)}$.

Alternatively: You may know that a group is solvable iff it has a normal subgroup $N$ so that $N$ and $G / N$ are both solvable. This also gives a proof.