Results 1 to 2 of 2

Math Help - homomorphism

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    166

    homomorphism

    Show that if H is any group and h is an element of H with h^n = 1, then there is a unique homomorphism from Z_n = <x> to H such that x --> h.

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

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by dori1123 View Post
    Show that if H is any group and h is an element of H with h^n = 1, then there is a unique homomorphism from Z_n = <x> to H such that x --> h.
    define f: <x> \longrightarrow H by f(x^k)=h^k, \ \forall k. it's obvious that f is a homomorphism and f(x)=h. you only need to prove that f is basically well-defined:

    if x^k=1, then k=mn, for some integer m. thus: f(x^k)=h^k=(h^n)^m=1. hence f is well-defined. this proves the existence of such homomorphism.

    now suppose g: <x> \longrightarrow H is another homomorphism with g(x)=h. then for all k: \ g(x^k)=(g(x))^k=h^k. hence g=f, which proves the uniqueness.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: April 19th 2013, 04:05 PM
  2. Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 15
    Last Post: June 25th 2011, 07:45 AM
  3. Is this a homomorphism?
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: August 27th 2010, 02:10 AM
  4. homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 21st 2009, 10:38 PM
  5. Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 15th 2009, 03:17 PM

Search Tags


/mathhelpforum @mathhelpforum