I apologise if this is a rather basic question- I am relatively new to group theory and geometry/topology etc. I was reading about the Euclidean Group, which Wikipedia defines as being...
the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean motions.
From what I am aware, isometries include translations, rotations and reflections. However later on in the Wiki article it talks about dimensionality and degrees of freedom:
The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.
I was confused as to why reflections did not factor in here, although intuitively I didn't see how a reflection could be a degree of freedom either. There just seems to be something different about reflections compared with translations and rotations, although I'm not sure what (not very mathematical, I know)
After reading about Classification of Euclidean plane isometries on Stack exchange, one of the answers described how you could describe any isometry by a translation, rotation and reflection by considering three points: use a translation to map one point onto its image, a rotation to map the second point to its image, and finally a reflection to match the third. From this I can see how once the translation and rotation are used to match two points, the reflection used for the third is entirely determined by the isomatry- there is no more freedom in choosing the reflection, if you have used a translation and rotation to specify the mapping of two of the points. For an isometry, the final step is fixed.
This kind of makes sense to me, although I am not sure if it is the best way, or even a correct way, of thinking about this.
Also, the answer also stated that
A non-zero translation and a non-zero rotation together are simply a rotation around some other center. A rotation followed by a reflection is a reflection in some different line. (MvG)
So it seems that all combinations of translations and rotations can be described by a single reflection? So reflections 'hide' all of the degrees of freedom? I think this is related to the final point, but I admit I do not quite understand it. Perhaps some of my confusion regarding transformations and degrees of freedom is that I find myself veering towards thinking of the Euclidean group as consisting of transformations as opposed to isometries, and it seems that these are not the same (in this link, it asks about isometries being a subgroup of the reflection group)