Thread: groups

I need help on the following problem:
Let a be the rotation about the origin in the plane and let p be the reflection in the x-axis. Show that pap-1=a-1.
Thanks

Originally Posted by thomas_donald

I need help on the following problem:
Let a be the rotation about the origin in the plane and let p be the reflection in the x-axis. Show that pap-1=a-1.
Thanks

you may use matrix to explain it.
the rotation can be represented as $\displaystyle a=\left(\begin{array}{cc}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right )$
the reflection can be represented as $\displaystyle p=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$
So $\displaystyle pa=\left(\begin{array}{cc}\cos\theta&-\sin\theta\\-\sin\theta&-\cos\theta\end{array}\right)$
$\displaystyle papa=(pa)^2=\left(\begin{array}{cc}1&0\\0&1\end{ar ray}\right)=I$
Thus $\displaystyle pap=a^{-1}\Rightarrow pap^{-1}=a^{-1}$since $\displaystyle p=p^{-1}$

Thank you very much for your help.