# Natural Group Homomorphism

1. Apr 2, 2005

### ti89fr33k

What is it, and can you give me a few examples of how its used?

Thanks,
Mary

2. Apr 3, 2005

### matt grime

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

3. Apr 3, 2005

### Hurkyl

Staff Emeritus
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.

4. Apr 4, 2005

### mathwonk

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.