The mathematical concept of a group is a bit different from the sort of practical math that is most commonly taught, such as algebra, trigonometry and calculus, but one of the great discoveries (of the 18th century) when it comes to describing the possible types of symmetry that can be found in isolated objects (point groups) or objects in a lattice (space groups). Returning to the operators For our purposes, we can consider these operations as the symmetry actions that will determine the positions of atoms in our unit cell.
Groups are built from operators, which for our purposes we will consider as giving us the coordinates for duplicate copies of atoms at other locations, as we saw before. To give this important mathematical concept a very brief overview, the key mathematical requirements of a group are that every group must have an identity operation and that group must be complete, meaning that any operator in the group if performed on another operator in a group will generate an operator that is also contained in the group. This means that all permutations of each operator combined generates an operator in the group. This also requires that there must be an inverse operator for every operator in the group.
We will write the identity operation as x,y,z, meaning that an atom with coordinates (x,y,z) is placed at those coordinates. The simplest group has only this operation and we can see that this group is complete, meaning that any repeated set of operators from the group, in any order, will only create another operator in the group. Thus, a single operator, x,y,z, creates our simplest group, since x,y,z operated on x,y,z gives us the original operator back. This, combined with a translation operators from an infinite lattice, is labeled by crystallographers as P 1 using what is known as the Hermann-Mauguin space group notation.
If we consider the mirror plane operator we saw before, x,y,-z, and combine that with x,y,z, we create a second group with just two elements, since x,y,-z operated on itself gives x,y,z and x,y,-z operated on x,y,z gives back x,y,-z. Combined with a lattice, these two operators will generate a space group P m using short-form Hermann-Mauguin notation (or P 1 1 m in long-form Hermann-Mauguin notation, where the 1 indicates there is no symmetry along the aor b axes.) Some head-scratching about mirror planes will show that a lattice must have an axis that is at right angle to the mirror plane (if not lattice points will not be reproduced at the correct positions), so P m is only consistent with a monoclinic unit cell. When I say only consistent with monoclinic, this does not preclude a unit cell that appears to be of higher symmetry, for example it could have all three cell axes appear as 90° within arbitrarily small experimental errors, but only one of these angles is constrained by symmetry to be 90° and thus the cell would not be considered to be orthorhombic unless additional symmetry is present that requires all angles to be 90°.
We can also take the P m space group and add an additional centering operation to create a new space group. Let’s add a B-center which means operator ½+x,y,½+z. When this is operated on x,y,-z we create fourth additional operator, ½+x,y,½-z. Using these four operators on each other, we generate no others. Thus, the space group B 1 1 m has four operators: x,y,z; x,y,-z; ½+x,y,½+z; and ½+x,y,½-z. Note that we used only two symmetry operations (the mirror plane and centering) to generate these four operators. We call those two operators the generators because those two operators can be used generate the entire group and those generators are thus included in the Hermann-Mauguin name. It is also worth noting that Volume A of the International Tables for Crystallography does not include B 1 1 m as one of the 230 space groups, but a closer look at the volume one can find B 1 1 m as one of the non-standard settings for space group A 1 m 1. The 230 three dimensional unique space groups tabulated in the International Tables for Crystallography, Volume A only reflects only the standard settings for space groups. Just as the International Tables lists B 1 1 m as one of the non-standard settings for space group A 1 m 1, one could also consider F centering or I centering with a mirror plane and generate other versions of space groups. There are a large number of non-standard space groups. These can always be reduced or transformed to one of the 230 standard space groups, but there can be good reasons why use of a non-standard space group can be very convenient, most commonly because it allows direct comparison of coordinates between related structures. As we will see, GSAS-II does allow you to use these non-standard settings, and they can be quite useful.
We saw before that the combination of operations -x,-y,-z and -x,y,½+z generates x,-y,½+z. Thus, we know that -x,-y,-z and -x,y,½+z alone cannot form a group, but you can convince yourself that the four operators, x,y,z, -x,-y,-z, -x,y,½+z and x,-y,½+z do constitute a group.
When considering classes of space groups, one must consider that a hexagonal cell can host a structure with a 6-fold (or \(\bar {6}\)) rotation axis or a 3-fold (or \(\bar {3}\)). The later are considered trigonal space groups and the former hexagonal. Since rhombohedral symmetry can also be expressed in a hexagonal cell, they are labeled as a special centered case within the trigonal space groups and are given an R prefix.