Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Natural Group Homomorphism

  1. Apr 2, 2005 #1
    What is it, and can you give me a few examples of how its used?

    Thanks,
    Mary
     
  2. jcsd
  3. Apr 3, 2005 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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
     
  4. Apr 3, 2005 #3

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  5. Apr 4, 2005 #4

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Natural Group Homomorphism
  1. Group homomorphism (Replies: 7)

  2. Group homomorphism (Replies: 4)

Loading...