I'm floundering with these two terms, and struggled to follow the responses given to similar questions elsewhere.
Background: I just about understand classes and groups, binary operations, and injection/bijection/surjection. However I struggle with the definitions of homomorphism and isomorphism as a 'structure-preserving mapping between two structures of the same type.' I haven't yet gotten into topology, which I note because some other answers were given in these terms.
Perhaps an example will illustrate my difficulty:
Say we have a function F(x) which takes the natural log of x, and in our example converts group A into group B.
Set A consists of 2, 3, and 6, while set B includes ln2, ln3, and ln6.
We note that 2*3 produces 6 in the first set, while in the second set we can do the same thing using addition instead. So let's say multiplication is the binary operation in the first group, and addition the binary operation in the second group.
My first question here: what is 'structure'? Is it the fact that we can link the same combinations of elements within the respective groups using its binary operation, even if the actual operation is different in each one? Which differences between the two groups would prevent the function being 'structure-preserving' and which others would be merely... superficial?
Second, what exactly are isomorphism/homomorphism a property of? Are they used to describe one or both groups, or the function, or the binary operation, or the elements? I have seen elsewhere the example of 'log(xy) = logx + logy' as satisfying homomorphism as long as the first set is limited to positive reals. This would seem to make homo/isomorphism a property of both the original group and the transformation. Is that the best way to think about it?
(I'm going to conclude optimistically that 'mapping' probably refers to the function.)
Third, how does this relate back to injective/bijective/surjective? I'm very used to the idea that all functions can be done the opposite way and am struggling with the idea that a transformation might not be reversible, which may be the barrier to my understanding.
If you're able to address the above in a fairly self-contained way without introducing too many additional concepts I would be very grateful.