Abelian Group of Prime-power Order is Product of Cyclic Groups/Corollary

Theorem

Let $G$ be an abelian group of prime-power order.

Then $G$ can be written as an internal direct product of cyclic groups of prime-power order.

Proof

From Lagrange's Theorem, we see that in the proof of Abelian Group of Prime-power Order is Product of Cyclic Groups, also $K$ satisfies the induction hypothesis.

The result follows as $\left\langle{a}\right\rangle$ is cyclic and of prime-power order.

$\blacksquare$