homomorphisms and binary algebraic structures What exactly does it mean to say a map is "structure preserving"? And what is the difference between this and a isomorphism?  
f:G -> H. 
f(g1 ∗ g2) = f(g1) ∗′ f(g2) . is the definition of a homomorphism. Which seems to say that you can transfer the binary algebraic structure in G to a binary algebraic structure in H. But intuitively, what does that really mean? I have no taken that many math classes, so if you could keep the examples you may use to as basic and primitive as possible, that would be terrific! 

A structure-preserving map $f:G\rightarrow H$ is one that "doesn't care" where you do the operation. For example, if we had groups $G,H$ with operations $\cdot, *$, then a homomorphism is structure-preserving in the sense that if we have elements $g,g' \in G$ and corresponding $f(g),f(g') \in H$, then it doesn't matter whether we do:


*

*Multiply $g\cdot g'$ (in $G$)

*Take the image of the product, $f(g\cdot g')$
Or the other way:
1'. Take the images $f(g), f(g')$
2'. Multiply the images $f(g)*f(g')$ (in $H$)
Either way, we arrive at the same value: that is, $f(g\cdot g') = f(g)*f(g')$, which is exactly the definition of a homomorphism.
Homomorphisms differ from isomorphisms insofar as the latter requires $f$ to be bijective; that is, that there is also an inverse map $f^{-1}:H \rightarrow G$ that is also a homomorphism.
