What is it, and can you give me a few examples of how its used?
A homomorphism is simply a map between groups that respects the groupiness of the groups. (I don't know why Natural is there, that has a specific category theoretic definition that isn't necessary at this stage.)
I don't follow what you mean by "used". I could tell you abuot the properties of homomorphisms that are important. We just find them useful. I mean what's the "use" of a surjection, and injection, addition, division. Perhaps if you explain waht you wanted to know in terms of how you "use" those then we could say more. 1
Maybe by "natural" he means something like the projection G --> G/K?
One example you've probably used quite a bit is the homomorphism from Z to Zp that maps an integer to the corresponding integer mod p.
heres a little example comprising both matt's categorical point and hurkyls example.
suppose you have a group map from Z to a group G and suppose that n goes to the identity e. then there is an induced map from Z/n to G such that the composition
Z-->Z/n-->G equals the original map Z-->G.
this is a naturality property of the map Z/n-->G.
Another one is that if G-->H is another group homomorphism, then n will still go to e under the composition Z-->G-->H, and the natrual map Z/n--H will equal the composition Z/n-->G-->H.
the fact that the construction of the map Z/n-->(anything), [factoring the map Z-->(anything)], behaves well under composition, is the categorical naturality property.
It is so natural that I did not bother to check it here, it just has to be true.
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