Results 1 to 7 of 7
Like Tree2Thanks
  • 1 Post By HallsofIvy
  • 1 Post By Plato

Thread: group homomorphism examples with comlpex numbers

  1. #1
    Newbie
    Joined
    Oct 2016
    From
    n ireland
    Posts
    8

    group homomorphism examples with comlpex numbers

    I think i get how to prove a homomorphism just when they add in complex numbers i panic!
    I need to prove whether the following are a homomorphism

    Complex numbers under addition
    a) sin(Rez)
    Complex numbers under multiplication
    b) e^z
    c) z/z
    Any help would be very gratefully appreciated.
    Many thanks
    Attached Thumbnails Attached Thumbnails group homomorphism examples with comlpex numbers-question.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,117
    Thanks
    2803

    Re: group homomorphism examples with comlpex numbers

    Well, do you know the definition of "homomorphism"? You say "I think i get how to prove a homomorphism" so I assume you do. You prove that a function from one group to another by showing that each point in the definition is true. For a function, f, from one group, A, to another, B, you must prove
    For any x, y in A, f(x* y)= f(x)* f(y) where "*" is the group operation.
    It is also true that f(0_A)= 0_B or f(1_A)= 1_B where 0_A, 0_B, 1_A and [tex]1_B[/tex[ are the additive and multiplicative identities of A and B and that [tex]f(x^{-1})= (f(x))^{-1} but, if I remember correctly those can be derived from the first two requirements so only those two need be shown.

    But your exercises are not complete- you give only a single group. In the second problem you say
    "Complex numbers under multiplication
    b) e^z"

    If you mean from the set of complex numbers, with multiplication as the operation to the set of complex numbers, with multiplication as the operation, this is not a homomorphism. But if you mean from the set of complex numbers, with multiplication as the operation to the set of complex numbers, with addition as the operation, then it is. That is because  (e^{z_1})(e^{z_2})= e^{z_1+ z_2}.
    Last edited by HallsofIvy; Mar 12th 2017 at 12:38 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2016
    From
    n ireland
    Posts
    8

    Re: group homomorphism examples with comlpex numbers

    From complex addition to complex addition, for part a
    Should i be using: Sin (Rez)=Sin(x)
    Then: Sin (x1+x2) Which wouldn't be a homomorphism according to angle sum and difference identities. Or am i completely on the wrong trail of thought.
    Many thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,117
    Thanks
    2803

    Re: group homomorphism examples with comlpex numbers

    That is correct. The first is not a homomorphism. But you have not addressed the more important point- these functions are from what group to what group?
    Thanks from smclaughlin78
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,164
    Thanks
    2609
    Awards
    1

    Re: group homomorphism examples with comlpex numbers

    Quote Originally Posted by HallsofIvy View Post
    you have not addressed the more important point- these functions are from what group to what group?
    I think that the attachment in the OP does just that.
    It says $\phi_1:\mathbb{C}\to\mathbb{C}$; $\phi_2:\mathbb{C}^*\to\mathbb{C}^*$ and; $\phi_3:\mathbb{C}^*\to\mathbb{C}^*$
    But it does not say what $\mathbb{C}^*$ is (multiplication ?)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Oct 2016
    From
    n ireland
    Posts
    8

    Re: group homomorphism examples with comlpex numbers

    In this question, C∗ is the group of non-zero complex numbers under multiplication,
    and C is the group of all complex numbers under addition.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,164
    Thanks
    2609
    Awards
    1

    Re: group homomorphism examples with comlpex numbers

    Quote Originally Posted by smclaughlin78 View Post
    In this question, C∗ is the group of non-zero complex numbers under multiplication,
    and C is the group of all complex numbers under addition.
    $\phi_2:\mathbb{C}\to\mathbb{C}$ defined by $z\mapsto \sin(\Re(z))$
    You seem confused about the fact that $\sin(\Re(z))= \sin(x)$ As you know $\sin(z)$ is not additive.

    $\phi_3:\mathbb{C}^*\to\mathbb{C}^*$ defined by $z\mapsto \exp(z)$
    $\exp(z)\cdot\exp(w)\ne\exp(z\cdot w)$


    $\phi_4:\mathbb{C}^*\to\mathbb{C}^*$ defined by $z\mapsto \dfrac{z}{\overline{z}}$ Can you show that $\overline{z\cdot w}=\overline{z}\cdot\overline{w}~?$
    Thanks from smclaughlin78
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. homomorphism from a group Z36 to a group of order 24
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Mar 23rd 2012, 09:35 AM
  2. Group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 5th 2010, 01:24 AM
  3. group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 24th 2010, 06:19 PM
  4. group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: Jan 19th 2010, 09:05 PM
  5. Homomorphism of P group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Nov 18th 2008, 09:07 PM

/mathhelpforum @mathhelpforum