Factors of Composition Series for Prime Power Group

Theorem

Let $G$ be a group such that $\order G = p^n$ 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:

$\set e = G_0 \subset G_1 \subset \ldots \subset G_n = G$

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

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

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

The result follows from Condition for Composition Series.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 74$