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Math Help - 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

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

    The 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

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

    The R should be the symbol representing the real numbers but I don't know how to put that in.
    You want to find, A,B,C so that,
    \begin{bmatrix}A&B\\C&0 \end{bmatrix} \begin{bmatrix} a & b \\ c & 0 \end{bmatrix} = \begin{bmatrix} a & b \\ c & 0 \end{bmatrix} <br />
\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

    \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)

     <br />
\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

    aB+bC=Ab+Bc

    which gives

    (a-c)B-(A-C)b=0

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

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