# group of rigid motions vs. symmetry group

I'm slightly confused when it comes to terminology, I am following Judson's Abstract Algebra and came across the following:

"The group of rigid motions of a square consists of 8 elements [...]" it then goes on to list them as the 4 rotations, 2 reflections and the last two combinations of rotations and reflexions.

But immediately afterward, it says that the group of rigid motions of a cube contains 24 elements (6 sides times 4 rotations that preserve the side facing upwards).

It appears to me that it is using the same term (group of rigid motions) to mean both the symmetry group in the first case, but also the rotation group in the second case.

My understanding of the situation leads me to believe that the symmetry group of a cube should contain 48 elements, not just 24, if we include reflexions, as was the case for the "group of rigid motions of a square".

Any help would be really appreciated.

• One should either speak of orientation-preserving rigid motions of three-dimensional space to have a group of order 24 for the cube. Or perhaps orientation-preserving and 3D are implicitly understood both times - but that means we are allowed to move the 2D square in 3D and flip it over. Then again, why don't we allow moving the cube in 4D? – Hagen von Eitzen Oct 12 '18 at 22:16