I have two solvable groups $G_1$ and $G_2$ and wonder whether their direct product $G_1 \times G_2$ is solvable.
I can easily write subnormal series $G_1 \times G_2 \rhd G_1 \rhd \cdots$ and $G_1 \times G_2 \rhd G_2 \rhd \cdots$ and I know that this series has to have abelian composition factors if and only if G is to be solvable. So if $(G_1 \times G_2)/G_1$ and $(G_1 \times G_2)/G_2$ are abelian, my group is solvable. I know that these factors are abelian if $(G_1 \times G_2)$ is abelian but I suspect they are not in general.
What also bugs me is that composition series have to be the same lenght. So if $G_1 \times G_2$ is solvable and $G_1$ has more composition factors than $G_2$ it has to be somehow guaranteed that I always find enough normal subgroups to "squeeze" in between $G_1 \times G_2$ and $G_2$ in the series $G_1 \times G_2 \rhd G_2 \rhd \cdots$
Is this correct and can somebody offer some intuition here?