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Is there a simple algebraic proof?

Thanks!

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    yeah i checked geometrically, i am trying to solve it algebraically as well by using the fact that Mk composed with Tab = Tab composed with Mk where AB is parallel to the line of reflection k. Tab is a translation and Mk is a reflection across line k AND Mk composed with Tab is glide reflection.2012-12-12

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Let the glide reflection be $T$. Without loss of generality the reflection part of $T$ is reflection in the $y$-axis. Assume that the translation part is by the vector $(a,b)$.

The reflection takes $(x,y)$ to $(-x,y)$. The translation part now takes us to $(-x+a,y+b)$. So $T$ takes $(x,y)$ to $(-x+a,y+b)$.

Do it again. The reflection part takes $(-x+a,y+b)$ to $(x-a,y+b)$, and the translation part takes this to $(x,y+2b)$. So $T^2$ is translation parallel to the $y$-axis by $2b$.

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    great i see now, thanks!2012-12-12