If, A,B, C are finite abelian groups then prove
(i) A direct sum B isomorphic to A direct sum C implies that B is isomorphic to C
(ii) A direct sum A isomorphic to B direct sum B implies A isomorphic to B.
Use commutative diagrams:
$\displaystyle \begin{matrix}A\bigoplus B & \to & A\bigoplus C \\ \downarrow & \searrow & \downarrow \\ B & \to & C\end{matrix}$
You may need to make the diagrams bigger (include groups like $\displaystyle (A\oplus B) / (A \oplus \{e_B\})$, depending on how much detail you need).