I am currently taking an introductory abstract algebra course, and learning about group theory, etc. One thing we have covered is symmetry groups of simple geometric shapes in 2D, and to a cursory extent in 3D.
Our definition for the symmetries of a 2D group (i.e. a square) includes rotations ($r, r^2, r^3, \ldots$), a horizontal flip ($h, h^2$), and compositions of the two ($rh, r^2h, \ldots$)
In looking briefly at a 3D cube, we are asked to consider two opposite sides of the cube (along an axis; i.e. swap all vertices that are directly above or below each other), and whether there is any way to swap the locations of the vertices of the two sides with each other as a rigid transformation. My text then goes on to say that this is not possible without distorting the cube, and that such a transformation is not considered to be part of the symmetry group of a cube.
My question is, why is this? As the definitions for a 2D shape vs. a 3D shape seem to be inconsistent in dimensionality with each other.
That is; we cannot perform a horizontal flip in 2D as a rigid transformation; this can only be done in 3D. So a purist 2D symmetry set for a 2D shape would seem to include only the rotations. However, we lump in what is basically a 3D transformation into the set of symmetries for a 2D square, without really commenting on it.
Then, in the case of the 3D cube, I suspect (but admittedly have not proved) that such a swapping of two opposing sides' vertices might be possible in 4D (so I may just be blindly overgeneralizing). But we only include in the symmetry group those rotational symmetries that occur purely in 3D for this 3D shape. Any horizontal-flip-equivalent (as seen in the 2D square case), which might take place in 4D for a cube, is seemingly ignored. I would tentatively argue the same for a 4D shape having a 5D "horizontal flip", and so on.
Is there a reason for this seeming discrepancy in format beyond simple human convenience? Am I overlooking something obvious/non-arbitrary in how these symmetries are restricted? For a subject so fixated on clear generalizations, I found this to be a minor but glaring wrinkle in the ointment.
I hope I have been clear in my framing of my question; thanks for your consideration.