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Thread: Homomorphisms and kernals

  1. #1
    Forum Admin topsquark's Avatar
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    Homomorphisms and kernals

    Okay, I was working through some examples of groups and how to find homomorphisms and go from there to get the kernal of the group action.

    For example, I am working on $\displaystyle D_8$. In case you want it here's the multiplication table:

    $\displaystyle
    \begin{array} {c||c|c|c|c|c|c|c|c|}
    D_8 & e & r & r^2 & r^3 & s & sr & sr^2 & sr^3 \\
    \hline \hline
    e & e & r & r^2 & r^3 & s & sr & sr^2 & sr^3 \\
    \hline
    r & r & r^2 & r^3 & e & sr^3 & s & sr & sr^2 \\
    \hline
    r^2 & r^2 & r^3 & e & r & sr^2 & sr^3 & s & sr \\
    \hline
    r^3 & r^3 & e & r & r^2 & sr & sr^2 & sr^3 & s \\
    \hline
    s & s & sr & sr^2 & sr^3 & e & r & r^2 & r^3 \\
    \hline
    sr & sr & sr^2 & sr^3 & s & r^3 & e & r & r^2 \\
    \hline
    sr^2 & sr^2 & sr^3 & s & sr & r^2 & r^3 & e & r \\
    \hline
    sr^3 & sr^3 & s & sr & sr^2 & r & r^2 & r^3 & e \\
    \hline

    \end{array}
    $

    This is the group of symmetry operations on a square. Defining the points of the square by the coordinates (1, 0), (0, 1), (-1, 0), and (0, -1). r is the rotation of the square by 90 degrees and s is the reflection of the corners over the x-axis.

    The group has 5 elements of order 2 and a cyclic subgroup of order 4. It is non-Abelian.

    Now, I want to find a homomorphism $\displaystyle \varphi : D_8 \times H \rightarrow H$. (H is a set, not necessarily a group.) My problem is finding an H such that |H| is not 8 and $\displaystyle ker ( \varphi )$ is non-trivial. Just to be clear, my text gives the following definition: $\displaystyle ker( \varphi ) = \{ d \in D_8 ~ | ~ d \cdot h = h ~\forall ~ h \in H \} ~ \forall d \in D_8$, where $\displaystyle d \cdot h$ is a group action. Now, I know that if H is a subset of $\displaystyle D_8$ then $\displaystyle ker( \varphi ) = {e}$ is trivial so I have to look for a set H not contained in $\displaystyle D_8$ but I can't seem to find one. Yes, isomorphisms of $\displaystyle D_8$ exist but I'm specifically looking for $\displaystyle | H | \neq | D_8 |$ and I don't know how to construct such a set H.

    I thought I had this concept down years ago. I think I'm missing something simple...

    Any thoughts?

    -Dan
    Last edited by topsquark; Apr 30th 2018 at 10:23 AM. Reason: Better Greek letter
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  2. #2
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    Re: Homomorphisms and kernals

    How about $\phi: D_8 \to D_8 / \{s\}$? Then $\ker(\phi) = \{e,s\}$. You can also do $\phi: D_8 \to D_8 / \{r^2\}$ which would have $\ker(\pi) = \{e,r^2\}$. Then there's $\phi: D_8 \to D_8 / \{r \}$ which I think has $\ker(\phi) = \{e,r,r^2,r^3\}$.
    Thanks from topsquark
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    Forum Admin topsquark's Avatar
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    Re: Homomorphisms and kernals

    Quote Originally Posted by SlipEternal View Post
    How about $\phi: D_8 \to D_8 / \{s\}$? Then $\ker(\phi) = \{e,s\}$. You can also do $\phi: D_8 \to D_8 / \{r^2\}$ which would have $\ker(\pi) = \{e,r^2\}$. Then there's $\phi: D_8 \to D_8 / \{r \}$ which I think has $\ker(\phi) = \{e,r,r^2,r^3\}$.
    Thanks for the comments. I had a nagging feeling so I just looked up the definition of kernal online and doublechecked my text definintion. $\displaystyle ker ( \varphi ) = \{ d \in D_8 ~ | ~ d \cdot h = e \}$. Somewhere or another I copied the definition wrong and it all makes much better sense.

    Thanks again for the reply.

    -Dan
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