Thread: products

does reflections in any axis passing the origin and rotation commute?

i think they do commute and hence the group structure of both tgt should be a direct product but am not sure why indirect products are used in symmetries?

They do not commute. Draw a square, label the vertices, and try it out. Dihedral groups are not abelian.

Rotations and reflections do not generally commute.

For example, consider reflection in the x-axis in R^2, and counterclockwise rotation by 90 degrees. Then the vector (0,1) ends up in (1,0) upon reflection and then rotation, while it ends up in (-1,0) upon rotation and then reflection.