In a previous post about enantiomorph, I mentioned the affine space in which we locate ourselves for talking about the space groups. Staying in the affine space, we have 230 space groups in total, and if we do not care about the handedness, we have 219 inequivalent space groups in total. This is something I covered in details in that post. There, we encountered the notion of affine space, and in crystallography, we will be encountering a lot of such notions in abstract algebra. So, I will use this post to present a basic summary about those abstract algebra notions that are relevant to crystallography (and maybe a bit beyond).
In Ref. [1], it was stated that ‘In affine space — i.e., no defined origin — there are only 219 space groups …’. This may give us the intuition that it is because we stay in the affine space that we have 219 space groups. No, it is not – staying in affine space gives us 230 space groups and only when we do not care about the handedness will we reduce the number to 219. The statement there in Ref. [1] is very confusing.
Here below is presented the summary diagram about quite a few notions involved in abstract algebra. It is intended to be used a knowledge map, preventing us from getting lost before and after diving into details about any single notion involved in there.
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