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

Proving an Isomorphism

  1. Feb 23, 2005 #1
    How would I go about proving the following:

    If G has an element of order n, then H has an element of order n.

    I am not sure how to start, if I should some how go about proving one to one and onto.

    Help
     
  2. jcsd
  3. Feb 23, 2005 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Who knows, since you've not explained what G and H are.

    But, guessing you mean let G and H be isomorphic groups, show that G has an element of order n iff H does.

    Suppose f is an iso from G to H. Let x be in G, then, f(x^r)=f(x)^r, hence ord(f(x))<=ord(x). by symmetry ord(x)=ord(f(x)).
     
    Last edited: Feb 23, 2005
  4. Feb 23, 2005 #3
    Isomorphism

    Sorry, I left that part out.

    The whole problem states

    Assume that G and H are groups and that G and H are isomorphic
    Then prove the statement
    If G has an element of order n, then H has an element of order n.
     
    Last edited: Feb 23, 2005
  5. Feb 24, 2005 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Which is what I showed, albeit in a very quick fashion. Do you understand the proof?
     
  6. Feb 24, 2005 #5
    Isomorphism

    No, I do not understand the proof. I am taking group theory and I do not understand the concepts. Do you know what I can do to help me understand some of the concepts?
     
  7. Feb 24, 2005 #6

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    The concept is simply a definition.

    the order of an element is the smallest positive r such that x composed with itself r times is the identity

    a group isomorphism is a structure preserving map f(xy)=f(x)f(y)

    so it follows f(x^r)=f(x)^r

    if x^r=e, the identity, then f(x)^r = e, so if r is minimal and positive such that x^r = e then f(x) has order at most r. So by symmetry, with g the inverse iso to f, it follows they must be equal.

    you need to think about it. it shouldn't be instantly obvious, it'll take time to understand, but it's supposed to
     
  8. Feb 24, 2005 #7
    Isomorphism

    Thank you, your explanation of the proof helped.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Proving an Isomorphism
  1. Proving isomorphism (Replies: 12)

  2. Prove not isomorphic? (Replies: 2)

Loading...