Factors of Solvable Group are Prime

Theorem

Let $G$ be a solvable group.

Let $\HH$ be a composition series of $G$.

Then all factor groups of $\HH$ 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.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 75$. Solvable Groups