I have recently met this very interesting problem in my Group theory course:
Let $ p $ be a prime number and $ 1 \leq n $ is a natural number such that G is the Abelian p-group $ G = Z_{p^n}=Z/(p^n) $. We are to show that G cannot be written as a direct sum of non-trivial proper subgroups of G.
To be honest I have no real idea here. I thought perhaps induction on n but then again I could not really proceed. I really need the help and appreciate it.
If $G$ is a direct sum of non-trivial subgroups, then there exist non trivial subgroups $A$, $B$ of $G$ such that $A\cap B=0$.
Show this, and show that in $Z/p^n$ all non-trivial subgroups intersect nontrivially.