Results 1 to 4 of 4

Math Help - Finding a specific homomorphism as an example of a theorem

  1. #1
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,903
    Thanks
    329
    Awards
    1

    Finding a specific homomorphism as an example of a theorem

    I am trying to construct an example of a theorem from my Algebra text.
    If f: G \to H is a homomorphism of groups and N is a normal subgroup of G contained in the kernel of f, then there is...
    At this point all I am trying to do is come up with a non-trivial example of the requirements of the theorem, that is to say a group N such that N is not simply the kernel of f. I started with D4 but can't think of a good group to make a homomorphism with. I might not be thinking big enough...

    Thanks!
    -Dan
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    Try mapping Z to Z with the map n -> mn, for some fixed m. Then let N=kmZ, k>1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,903
    Thanks
    329
    Awards
    1
    Quote Originally Posted by Tinyboss View Post
    Try mapping Z to Z with the map n -> mn, for some fixed m. Then let N=kmZ, k>1.
    I'll give that a look. Thanks!

    -Dan
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,386
    Thanks
    752
    you can use D4, it actually provides a good example. in this case, let H be a cyclic group of order 2. the idea is we map all the rotations to e in H, and all the reflections to x ≠ e in H.

    now the kernel of such a homomorphism is the rotations {1,r,r^2,r^3}, and it turns out that we can take N to be the normal subgroup of D4: {1, r^2}.

    (the normality of this can be seen from the fact that this is the center of D4 (i think in some other post you called "r" C1).).

    the fact that f: D4-->H is a homomorphism can be summed up geometrically as "a reflection followed by another reflection is a rotation".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. finding a specific operator
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: January 15th 2012, 10:32 AM
  2. Finding a specific curve
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 4th 2011, 03:56 PM
  3. Replies: 8
    Last Post: October 20th 2011, 09:43 AM
  4. 2nd fund. theorem question on specific part
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 30th 2009, 06:16 AM
  5. Finding the specific digit of a large number
    Posted in the Algebra Forum
    Replies: 1
    Last Post: June 23rd 2009, 09:56 AM

Search Tags


/mathhelpforum @mathhelpforum