Give a two-sentence synopsis of the proof of Theorem 11.5. 

Data from Theorem 11.5.

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 Z_pr is indecomposable, for if Z_pr were isomorphic to Z_pi x Z_pj, where i + j = r, then every element would have an order at most p^max(i,j) r.