Centre of a group

**Nightfly** · December 9th 2008, 02:31 PM

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

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

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

Thanks.

**ThePerfectHacker** · December 9th 2008, 03:31 PM

Originally Posted by Nightfly

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

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

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,
$\begin{bmatrix}A&B\\C&0 \end{bmatrix} \begin{bmatrix} a & b \\ c & 0 \end{bmatrix} = \begin{bmatrix} a & b \\ c & 0 \end{bmatrix} \begin{bmatrix}A&B\\C&0 \end{bmatrix} \text{ for all }a,b,c$ .

**Nightfly** · December 10th 2008, 08:56 AM

Yeah, that's what I thought except I got

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

$ \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 $A,B,C$

**ThePerfectHacker** · December 10th 2008, 11:16 AM

Originally Posted by Nightfly

$aB+bC=Ab+Bc$

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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