# Group Theory-Group of Motions of plane

I have a doubt in Chapter 5, section 2 (Group Motions of the Plane) of Michael Artin. Page 159 has a paragraph which describes how computations in group $M$ of all rigid motions in a plane (translation, rotation, and reflection) can be done using symbols as given in the attached picture. While it is easy to understand the compositions ticked in green, I have not understood the compositions boxed in red. References 2.3, 2.4 from the book also has been added to the picture.

Would be grateful for any help in understanding these.

• Draw sketches for these compositions.. – Berci Mar 2 '18 at 8:13
• To convince yourself that the transformations are the same, you can just multiply the relevant matrices – rbird Mar 2 '18 at 8:15

$$\rho_\theta t_a = t_{a'}\rho_\theta$$
The parallelogram is just to show how $a' = \rho_\theta a$ is obtained: by rotating $a$. Using the same logic you can make sense of the other identity