Theorem 11.15

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**Statement:**

The finite indecomposable abelian groups are exactly the cyclic groups with order a power of a prime.

**Proof:**

Let G be a finite indecomposable abelian group. Then by Theorem 11.12, G is isomorphic to a direct product of cyclic groups of prime power order. Since G is indecomposable, this direct product must consist of just one cyclic group whose order is a power of a prime number.

Conversely, let $p$ be a prime. Then $\mathbb{Z}_{p^r}$ is indecomposable, for $\mathbb{Z}_{p^r}$ were isomorphic to $\mathbb{Z}_{p^i}$ X $\mathbb{Z}_{p^j}$, where $i+j=r$, then every element would have an order at most $p^{max(i,j)}<p^{r}$.