# Wallpaper Pattern: is this pmg?

I'm reading David R. Finston and Patrick J. Morandi's book Abstract Algebra: Structure and Application and in its last section 10.6 "the 17 wallpaper groups" page 179 I'm confused on what it gives as an example of $$pmg$$:

According to its description on pg 178, $$pmg$$ has point group $$D_2$$ , so it shall contains a rotation. Wikipage on pmg also says so:

"The group pmg has two rotation centres of order two (180°), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes."

But I couldn't find a rotation here in the pattern.

So, where did I miss?

If takes the red dot as the rotation centre, the two blue parts don’t match:

• @MarkBennet thx, good to know i didn't misunderstand it.. – athos Sep 27 '19 at 10:32
• If the grey tiles crown up are all different from all the grey tiles crown down then no half turn can take any grey tile into another, so no half turn symmetries can exist. I reckon that is a good spot. (autocorrect typo corrected) – Mark Bennet Sep 27 '19 at 12:51

You can rotate at the intersection points of $$4$$ tiles by $$180$$ degrees. There are two kinds of intersections points depending on whether say the upper right is white or grey.