Searching for an alternative definition of group homomorphism I am searching for an alternative definition of a group homomorphism using only their impact on subgroups. I thought of something like

Let $G$, $G'$ be groups. A map from $G$ to $G'$ is called homomorphism of groups if the image of every subgroup of $G$ is a subgroup in $G'$.

I guess this definition wouldn't work. It's maybe too weak to be a homomorphism.
Does anyone have an idea?
 A: Others have pointed out the trouble with this definition. This is the way one learns anyway. Try to see why this definition and not this gives a clear and critical understanding.
Let me point out another example where your suggested alternative definition fails: Take a group which is NOT abelian, say symmeric group on $n$ letters, . $n>3$. Consider the function $x\mapsto x^{-1}$. As a function from a group to itself, it satisfies your condition, image of any subgroup is a subgroup (in fact it is the same subgroup). However it is not true in general that $(xy)^{-1}= x^{-1}y^{-1}$. So it is not a homomorphism.
A: This definition is far too weak! you need compatibility with the group structure, which gets completely forgotten by your definition, for example, consider the following map: $$\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/q\mathbb{Z}\\
x \mapsto [x]$$ where we interpret $\mathbb{Z}/p\mathbb{Z} $ as $\{0,1,...,p-1\}$.
Now if $q \le p$ this is would be by your "definition" a "groupmorphism", but all the properties of the elements, like order gets messed up, and you lose all control!
