In the chapter I am reading on wallpaper groups, it outlines proofs that all the wallpaper groups are not isomorphic and hence different. But I do not fully see why what they are saying is true.
For example, it says that $p4mm$ and $p4gm$ are not isomorphic because the 4-fold rotation $(\textbf{a}, A_{\pi/2})$ in $p4gm$ cannot be written as the product of two reflections in $p4gm$, whilst each 4-fold rotation in $p4mm$ can be factorised as two reflections in $p4mm$.
* $A_{\pi/2}$ is the rotation matrix of order 4
Is someone able to show the steps it takes to conclude this?
The notation is that used by Armstrong in Groups and Symmetry.