Results 1 to 3 of 3

Math Help - Isomorphic groups

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    166

    Isomorphic groups

    Let f: G --> H be an isomorphism, G and H are groups.
    I've already showed that for every x in G, |f(x)| = |x|. (|x| = order of x, the smallest positive integer such that x^n is the identity element)

    How to show that any two isomorphic groups have the same number of elements of n, where n is any positive integer?

    Please help. Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2008
    Posts
    1,092
    If two groups are isomorphic, then they have the same cardinality. In order to show that two groups are isomorphic, you must display the isomorphism, that is, the conversion chart that shows how the elements of one group "translate" to the elements of the other group, and that every group operation checks out. Simply by listing the elements of each group, you can show that the groups have the same cardinality. It is also not difficult to show that both groups have an infinite number of members, if that is indeed the case.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by dori1123 View Post
    How to show that any two isomorphic groups have the same number of elements of n, where n is any positive integer?
    Because if two groups are isomorphic there is a bijection between them. Therefore, if one group has n elements the other must automatically have n elements because of the bijection.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Isomorphic groups.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 12th 2010, 02:56 AM
  2. isomorphic groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 14th 2009, 10:26 PM
  3. Isomorphic groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 22nd 2009, 08:07 PM
  4. Non isomorphic groups
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: July 20th 2008, 11:21 AM
  5. Isomorphic Groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: July 18th 2008, 07:10 AM

Search Tags


/mathhelpforum @mathhelpforum