Results 1 to 2 of 2

Thread: Homomorphisms, Integers mod 6 to Symmetry gp

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    6

    Homomorphisms, Integers mod 6 to Symmetry gp

    There is a unique homomorphism $\displaystyle \theta: \mathbb{Z}_{6}\rightarrow S_{3} $ such that $\displaystyle \theta ([1]) =(1 \ 2 \ 3).$ Determine $\displaystyle \theta ([k])$ for each $\displaystyle [k] \in \mathbb{Z}_{6}$. Which elements are in $\displaystyle Ker \theta $?

    I don't understand how that single condition can be used to determine the rest of the $\displaystyle \theta ([k])$. It doesn't seem like enough information.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Blue Calx View Post
    There is a unique homomorphism $\displaystyle \theta: \mathbb{Z}_{6}\rightarrow S_{3} $ such that $\displaystyle \theta ([1]) =(1 \ 2 \ 3).$ Determine $\displaystyle \theta ([k])$ for each $\displaystyle [k] \in \mathbb{Z}_{6}$. Which elements are in $\displaystyle Ker \theta $?

    I don't understand how that single condition can be used to determine the rest of the $\displaystyle \theta ([k])$. It doesn't seem like enough information.
    If $\displaystyle \theta (1) = (123)$ then $\displaystyle \theta (2) =\theta(1+1) = (123)^2$.
    And $\displaystyle \theta(3) = \theta(1+1+1) = (123)^3$.
    And so on ...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. All homomorphisms between S3 and Z6
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Oct 8th 2011, 04:47 PM
  2. homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 5th 2011, 06:19 PM
  3. Replies: 7
    Last Post: Aug 3rd 2010, 01:31 PM
  4. Matrix of integers whose inverse is full of integers
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Mar 19th 2010, 02:02 PM
  5. Replies: 4
    Last Post: Feb 24th 2008, 03:08 PM

Search Tags


/mathhelpforum @mathhelpforum