Factors of Solvable Group are Prime

Theorem
Let $G$ be a solvable group.

Let $\mathcal H$ be a composition series of $G$.

Then all factor groups of $\mathcal H$ must be prime.

Proof
By definition, a composition series is a normal series whose factor groups are all simple.

A solvable group, by definition, is one which has a composition series whose factor groups are all cyclic.

From Cyclic Group is Simple iff Prime, it follows that all the factor groups of a composition series of a solvable group must all be prime.