How to classify all the abelian groups with finite exponent?

Question

Let $A$ be an abelian group, the exponent exp$A$ is the least natural number $n$ (if exists) such that $nA=0$ or $+\infty$. The question can be reduced to the case exp$A=p^n$ for a certain prime number $p$ easily. But then I have no ideal. I'm not very sure if such a group must be a direct sum of cyclic groups, just like the finitely generated case. In particular, I'm not very sure if a infinite direct product of $C_{p^n}$ is direct sum of cyclic groups.

Answer 1

You can find in the first few pages of Irvin Kaplansky 's book Infinite abelian groups a proof of the statement that an abelian group of finite exponent is a direct sum of cyclic groups.