# What does it mean to assign group operations to distinct sets for space groups?

I am trying to understand space groups in crystallography. In International tables for crystallography, for a nonsymmorphic space group, they list some symmetry operations. 8 of them are listed under the (0,0,0)+ set and 8 in the (1/2, 1/2, 1/2)+ set. What does this mean? Are there 16 operations in total? How do the sets differ?

Edit: I find similar notation for symmorphic space groups as well. There are some space groups with only one set and some with two or more, and I don't understand what determines the number of sets.

• +1. Welcome to the site and thank you for contributing your question here!!! We hope to see much more of you in the future! I pinged someone who often answers group theory questions, to draw their attention to this question, but they said the question is a bit too vague and they would be able to answer if you give a concrete example. Can you give a specific example and elaborate in more detail? Nov 27, 2020 at 21:49
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## 1 Answer

When your lattice is primitive you have only the (0,0,0)+ set; when your lattice has some kind of centering (body- or face-centering) other sets are present, such as (1/2, 1/2, 1/2)+ or (1/2, 1/2, 0)+

It's not clear to me what you write. In the first page of the Internationl Tables you find all the symmetry operations that are listed with Roman numerals (1),(2),.... under the heading "Symmetry operations".

Then in the following page under the heading "Positions" you find the differnt Wyckoff sites. The first one represents the general position.

Consider the Space Group n°40 - Imm2. It has 8 symmetry operations, 4 for the (0,0,0)+ set and 4 for the (1/2,1/2,1/2)+ As you have 8 symmetry operations, you will have 8 different coordinates for the general position (8e). The Table gives you only the 4 coordinates referring to the (0,0,0)+ set; so the position x,y,z is given by the operation (1) (identity) and for this reason you read (1)x,y,z. the operation (2) is the 2-fold rotation along z, and the corresponding coordinates are -x,-y,z (in other words how the point in x,y,z is moved in a new position after applying the 2-fold rotation). and so on...

The 4 remaining coordinates are obtained by simply adding (1/2,1/2,1/2) to each of the 4 above ones.

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