Let $G = H \oplus K$ be abelian group. Now, I follow the definition of direct sum from Wikipedia which is
Now, we choose two element (both of them are not identity) $h \in H$ and $k \in K$ ($h, k \neq e_{G}$). When I tried to think about the element $h \cdot k$, then a contradiction seems to appear.
Let's assume the element $h \cdot k \in H \Rightarrow h^{-1}\cdot h \cdot k \in H \Rightarrow k \in H$. Contradiction.
Similarly If assume the element $h \cdot k \in K \Rightarrow k^{-1} \cdot h \cdot k \in K \Rightarrow h \in K$. Contradiction.
The element $h \cdot k $ doesn't seem to belong anywhere. What am I doing wrong?
Those come from $G$ itself.
Note that $H$ and $K$ are subgroups of $G$. Direct sum does not try to take two random groups and create a new one, it's rather about making sure that the subsets you've taken actually form the original group.