Factors of Composition Series for Prime Power Group

Theorem
Let $G$ be a group such that $\left|{G}\right| = p^n$ where $p$ is a prime number.

Then $G$ has a composition series in which each factor group is cyclic of order $p$.

Proof
From Composition Series of Group of Prime Power Order‎, $G$ has a sequence of subgroups:
 * $\left\{{e}\right\} = G_0 \subset G_1 \subset \ldots \subset G_n = G$

such that $\left|{G_k}\right| = p^k$, $G_k \triangleleft G_{k+1}$ and $G_{k+1} / G_k$ is cyclic and of order $p$.

From Prime Groups are Simple it follows that $G_{k+1} / G_k$ is a simple group for all $k$.

From Group of Prime Order Cyclic it follows that $G_{k+1} / G_k$ is cyclic.

The result follows from Condition for Composition Series.