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Math Help - Quotient groups, isomorphisms, kernels

  1. #1
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    Quotient groups, isomorphisms, kernels

    Let G be the group { <br />
      \begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}<br />
      | a, b, c are in Z_p with p a prime}
    Then let K = { <br />
      \begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix}<br />
      | b in Z_p}

    The map P: G --> Z_p* x Z_p* is defined by
    P( <br />
      \begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}<br />
      ) = (a, c)


    (A). Prove the quotient group G/K is isomorphic to Z_p* x Z_p*
    (B). Find the orders |g|, |K|, and |G/K|

    my thoughts -
    (A) I've already shown that K is normal in G, and I know this would be true if K were the kernel of P, but I don't know how to show that.
    (B), I'm not sure -- is it |G|= 3| Z_p|, |K|=| Z_p|, and |G/K| = 2| Z_p|?

    Thank you so much for any help!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kimberu View Post
    Let G be the group { <br />
      \begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}<br />
      | a, b, c are in Z_p with p a prime}
    Then let K = { <br />
      \begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix}<br />
      | b in Z_p}

    The map P: G --> Z_p* x Z_p* is defined by
    P( <br />
      \begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}<br />
      ) = (a, c)


    (A). Prove the quotient group G/K is isomorphic to Z_p* x Z_p*
    (B). Find the orders |g|, |K|, and |G/K|

    my thoughts -
    (A) I've already shown that K is normal in G, and I know this would be true if K were the kernel of P, but I don't know how to show that.
    (B), I'm not sure -- is it |G|= 3| Z_p|, |K|=| Z_p|, and |G/K| = 2| Z_p|?

    Thank you so much for any help!
    What is \mathbb{Z}_{p{\color{red}*}}?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    What is \mathbb{Z}_{p{\color{red}*}}?
    I'm actually not sure - I assumed it was the group Z_p under multiplication.

    Edit: Sorry, I looked in my textbook and it says " Z_p*" denotes the elements in Z_p that have inverses under multiplication.
    Last edited by kimberu; April 13th 2010 at 06:51 PM. Reason: updated definition
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kimberu View Post
    I'm actually not sure - I assumed it was the group Z_p under multiplication.
    That's usually denoted \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}

    Anyways, if you prove that this P is an epimorphism it follows that K=\ker P and so G/K\cong \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times}\times\left(\mathbb{Z}/p\mathbb{Z}\right)^\times

    P.P.S. It makes sense that K\subseteq\ker P since if (a_{ij})\in K then a_{11}=a_{22}=1 and so P((a_{ij}))=(a_{11},a_{22})=(1,1) and the RHS is the identity in the codomain.
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  5. #5
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    Red face

    Quote Originally Posted by kimberu View Post
    Let G be the group { <br />
\begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}<br />
| a, b, c are in Z_p with p a prime}
    Then let K = { <br />
\begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix}<br />
| b in Z_p}

    The map P: G --> Z_p* x Z_p* is defined by
    P( <br />
\begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}<br />
) = (a, c)


    (A). Prove the quotient group G/K is isomorphic to Z_p* x Z_p*
    (B). Find the orders |g|, |K|, and |G/K|

    my thoughts -
    (A) I've already shown that K is normal in G, and I know this would be true if K were the kernel of P, but I don't know how to show that.
    (B), I'm not sure -- is it |G|= 3| Z_p|, |K|=| Z_p|, and |G/K| = 2| Z_p|?

    Thank you so much for any help!

    For the above to be true it must be that G:=\left\{\begin{pmatrix}a&b\\0&c\end{pmatrix}\;\;  ;\;a,b,c\in\mathbb{Z}_p\,,\,\,ac\neq 0\right\} ; Now it's clear the map P is a group epimorphism and its kernel is K since (1,1) is the unit element in the group \mathbb{Z}^{*}_p\times \mathbb{}Z^{*}_p , and for the orders:

    |G|=(p-1)p(p-1)=p(p-1)^2\,,\,\, |K|=p\,,\,\, \left|G\slash K\right|=(p-1)^2.

    Finally: yes, \mathbb{Z}^{*}_p=\left(\mathbb{Z}\slash p\mathbb{Z}\right)^{*}=\mathbb{Z}_p-\{0\}= the multiplicative group of the field \mathbb{Z}_p

    Tonio
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