TITLE: How can one visualize a homomorphic mapping.
QUESTION [10 upvotes]: It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily,  yet not fully understanding what a homomorphic map really is. Though, i find it much easier to visualize an isomorphic map.
I know the definition. But I'd like to understand it with a visual approach. Any help would be appreciated.

REPLY [2 votes]: Group and ring structures (on a set $A$) is about one respectively two functions 
$A\times A\rightarrow A$. 
The simple connection is that homomorphisms preserves structures as relations. The simplest case is graphs and graph homomorphism, where connected nodes maps on connected nodes: 
$x\rightharpoonup y\implies\varphi(x)\rightharpoonup \varphi(y)$ 
For functions $A\times A\rightarrow A$, $(a,b)\in A^2$ is "connected" with 
$c\in A$ if and only if $a\star b=c$, and therefore $(f(a),f(b))$ is "connected" to 
$f(c)$ if $f$ is a homomorphism. That is: 
$f(a)\star f(b)=f(c)=f(a\star b)$.
There is a lot of mathematical structures on sets, where homomorphisms maps "connected" elements on "connected" elements.