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

Isomorphic groups that have different properties?

  1. Aug 22, 2008 #1

    tgt

    User Avatar

    What are some properties apart from the actual names of the elements that differ between isomorphic groups?
     
  2. jcsd
  3. Aug 22, 2008 #2

    radou

    User Avatar
    Homework Helper

    Your question actually is - what, in general, are the differences between different groups?

    Elements and the defined binary operation.
     
  4. Aug 23, 2008 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    that's pretty much it. Two isomorphic groups (or fields or... pretty much anything isomorphic) differ only in the "labeling": what you name the elements and operations.
     
  5. Aug 23, 2008 #4

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    in practice it is sometimes a little more subtle than it seems.

    e.g. we usually give a cyclic group in the form (Z/n,+) but may not notice that this gives not only the group but also a distinguished generator, namely 1.

    the group( (Z/p)*, .) of units in the ring Z/p is also cyclic when p is a prime, but it may not be immediately clear what a generator is. moreover since there are many generators, (Z/p)* and no one is naturally distinguished, there is no completely natural way to choose an isomorphism between the groups (Z/p)* and Z/(p-1).

    i.e. in a sense, ({1,2,3,....,p-1}, . ) are just different names for ({0,1,2,...,p-2},+) but it is not clear which new nAME CORRESPONDS TO WHICH OLD NAME.

    e.g. when p = 7, (Z/7)* is isomorphic to Z/6, and the least generator is 3, perhaps it is natural to associate n with 3^n, for n=0,...,5, but I am not sure this generator is always best.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Isomorphic groups that have different properties?
  1. Isomorphic groups? (Replies: 3)

  2. Isomorphic groups (Replies: 5)

  3. Isomorphic groups (Replies: 12)

Loading...