I have the following problem:
Let $(G, +)$ and $(H, *)$ be groups. Prove that the projection maps $\pi_{1}: G×H\to G$ and $\pi_{2}: G×H\to H$ are homomorphisms.
My attempt:
We know by definition, that a homomorphism is a function $f: A \to B$ that satisfies the following condition: For all $a, b \in A$ it implies $f(a) * f(b) = f(a +b)$ where $(A, +)$ and $(B, *)$ are groups.
Let's prove that $π_{1}$ is a homomorphism.
Let $(a, b), (c, d) \in G × H$, then we have to prove that $\pi_{1}((a, b) ? (c, d)) = \pi_{1}((a, b)) + \pi_{1}((c, d))$
Here's where I have my question, how is define the operation $?$. By hypothesis, we now that the operations defined in $G$ and $H$ are $+$ and $*$ respectively, but never mention how the elements of the cartesian product $G×H$ are operated.
Hope you can help me :)
The operation $?$ is simply that of the direct product. See the comment by @SMM:
$$(g_1,h_1)?(g_2,h_2):= (g_1+g_2,h_1*h_2). $$
I suggest that you use a different symbol than $?$, like $\cdot$ given by
$\cdot$for instance.