What is a natural homomorphism from $G$ to $G\times H$? Let $G$ and $H$ be two groups. What would a "natural homomorphism" from $G$ to $G\times H$ look like?
My book mentions that one may assign natural homomorphisms from $G$ and $H$ to $G\times H$, but I don't understand the statement. Does it mean I map $g\to (g,e_H)$?
 A: Yes, the map $\varphi_G\colon G\to G\times H$ is just the one which maps the element $g\in G$ to the element $(g,e_H)$ where $e_H\in H$ is the identity element of $H$. The question of why this is a 'natural' map is a good one. Perhaps the most enlightening answer, though conceptually more complex, is that this is precisely the map which is given by the universal property of products in the category of groups $\mathbf{Grp}$ for the identity map on $G$ and the trivial map from $G$ to $H$.
In particular we have the following theorem.

Theorem. For any two groups $G$ and $H$, there exists a group $K$ and homomorphisms $\pi_G\colon K\to G$ and $\pi_H\colon K\to H$ such that for any other group $L$ and homomorphisms $f_G\colon L\to G$ and $f_H\colon L\to H$, there exists a unique homomorphism $f\colon L\to K$ with the property that $\pi_G\circ f=f_G$ and $\pi_H\circ f=f_H$.

This is known as the universal property of products, and you will probably have guessed by now that the group $K$ in the above definition is just the group $G\times H$ (or is isomorphic to it) and the homomorphisms $\pi_G$ and $\pi_H$ are the usual projections onto the factors of $G\times H$. It's because of this that many people take this to be the definition of the product, rather than just as a property which the product satisfies (this becomes an important interpretation when abstracting the concept of a product to other collections of 'objects' and 'maps')
Now, consider the homomorphisms $\mbox{Id}_G\colon G\to G$, the identity homormohpism, and $t\colon G\to H$, the trivial morphism mapping all elements to the identity element in $H$ - these are some of the only homomorphisms which we can guarentee exist from $G$ to $G$(resp. $H$) for arbitrary groups (we also have the trivial map $G\to G$ which would give rise to the trivial map $G\to G\times H$ via the universal property).
By the above theorem, there must exist a unique map $\varphi_G\colon G\to G\times H$ such that $\pi_G\circ \varphi_G=\mbox{Id}_G$ and $\pi_H\circ \varphi_G=t$. You can quite easily check that the map $g\mapsto (g,e_H)$ satisfies these conditions, and so this map arises 'naturally' almost directly from the defining characteristic of the product. Swapping $G$ and $H$ in everything above will give you the same for the natural map $H\to G\times H$.
