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Thread: Centre of a group

  1. #1
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    Centre of a group

    Could anyone help me please? I don't have any examples of how to work out the centre of a group, so I'm struggling. I keep trying to apply the definition but any way I try doesn't seem to work.

    Work out the centre of

    $\displaystyle T(2,R)=
    \{ \left(\begin{array}{cc}a&b\\0&c\end{array}\right); a,b,c $ elements of $\displaystyle R, ac \neq 0 \}
    $

    The $\displaystyle R$ should be the symbol representing the real numbers but I don't know how to put that in.


    Thanks.
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  2. #2
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    Quote Originally Posted by Nightfly View Post
    Could anyone help me please? I don't have any examples of how to work out the centre of a group, so I'm struggling. I keep trying to apply the definition but any way I try doesn't seem to work.

    Work out the centre of

    $\displaystyle T(2,R)=
    \{ \left(\begin{array}{cc}a&b\\0&c\end{array}\right); a,b,c $ elements of $\displaystyle R, ac \neq 0 \}
    $

    The $\displaystyle R$ should be the symbol representing the real numbers but I don't know how to put that in.
    You want to find, $\displaystyle A,B,C$ so that,
    $\displaystyle \begin{bmatrix}A&B\\C&0 \end{bmatrix} \begin{bmatrix} a & b \\ c & 0 \end{bmatrix} = \begin{bmatrix} a & b \\ c & 0 \end{bmatrix}
    \begin{bmatrix}A&B\\C&0 \end{bmatrix} \text{ for all }a,b,c$.
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  3. #3
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    Yeah, that's what I thought except I got

    $\displaystyle \left(\begin{array}{cc}a&b\\0&c\end{array}\right)\ left(\begin{array}{cc}A&B\\0&C\end{array}\right)=\ left(\begin{array}{cc}aA&aB+bC\\0&cC\end{array}\ri ght)$

    $\displaystyle
    \left(\begin{array}{cc}A&B\\0&C\end{array}\right)\ left(\begin{array}{cc}a&b\\0&c\end{array}\right)=\ left(\begin{array}{cc}aA&Ab+Bc\\0&cC\end{array}\ri ght)$

    so

    $\displaystyle aB+bC=Ab+Bc$

    which gives

    $\displaystyle (a-c)B-(A-C)b=0$

    but then I couldn't figure out how to get the values for $\displaystyle A,B,C$
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  4. #4
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    Quote Originally Posted by Nightfly View Post
    $\displaystyle aB+bC=Ab+Bc$
    This equation is satisfied for all $\displaystyle a,b,c\in \mathbb{R}, ac\not = 0$.
    If this works for any values then it works for $\displaystyle b=0$ and we get,
    $\displaystyle aB = Bc \implies (a-c)B = 0 \implies B=0$ since $\displaystyle a-c\not = 0$ for all $\displaystyle a,c$.
    But if $\displaystyle B=0$ then it forces $\displaystyle A=C$.

    Thus the center is the set, $\displaystyle \left\{ \begin{bmatrix} t&0\\0&t \end{bmatrix} : t\in \mathbb{R}^{\times} \right\}$.
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