Group Multiplication Table
A table constructed by selecting any two elements from a group and arranging their operation results (products) is called a group multiplication table (Cayley1 table) (also known as a multiplication table or product table). For example, the multiplication table of the point group \(\bar{1}\), which contains the identity transformation (\(1\)) and inversion (\(\bar{1}\)) as elements, is as follows. On this page, we proceed with the convention that the row element is multiplied from the left and the column element from the right.
| \(1\) | \(\bar{1}\) | |
| \(1\) | \(1\) | \(\bar{1}\) |
| \(\bar{1}\) | \(\bar{1}\) | \(1\) |
The multiplication table of the point group \(m\) (principal axis \(b\)) is as follows. Note that Seitz notation is used for the symmetry operation symbols (see here).
| \(1\) | \(m_{010}\) | |
| \(1\) | \(1\) | \(m_{010}\) |
| \(m_{010}\) | \(m_{010}\) | \(1\) |
The multiplication table of the point group \(\bar{4}\) (principal axis \(c\)) is as follows.
| \(1\) | \(2_{001}\) | \(\bar{4}^+_{001}\) | \(\bar{4}^-_{001}\) | |
| \(1\) | \(1\) | \(2_{001}\) | \(\bar{4}^+_{001}\) | \(\bar{4}^-_{001}\) |
| \(2_{001}\) | \(2_{001}\) | \(1\) | \(\bar{4}^-_{001}\) | \(\bar{4}^+_{001}\) |
| \(\bar{4}^+_{001}\) | \(\bar{4}^+_{001}\) | \(\bar{4}^-_{001}\) | \(2_{001}\) | \(1\) |
| \(\bar{4}^-_{001}\) | \(\bar{4}^-_{001}\) | \(\bar{4}^+_{001}\) | \(1\) | \(2_{001}\) |
The multiplication table of the point group \(222\) is as follows.
| \(1\) | \(2_{100}\) | \(2_{010}\) | \(2_{001}\) | |
| \(1\) | \(1\) | \(2_{100}\) | \(2_{010}\) | \(2_{001}\) |
| \(2_{100}\) | \(2_{100}\) | \(1\) | \(2_{001}\) | \(2_{010}\) |
| \(2_{010}\) | \(2_{010}\) | \(2_{001}\) | \(1\) | \(2_{100}\) |
| \(2_{001}\) | \(2_{001}\) | \(2_{010}\) | \(2_{100}\) | \(1\) |
The multiplication table has the property that when any single column or row is selected, every element appears exactly once.
Group Isomorphism
When an isomorphic mapping exists between two groups \(G, H\), the relationship between \(G, H\) is called isomorphism. An isomorphic mapping is a mapping by a function \(f\) that maps elements of \(G\) to elements of \(H\), satisfying the property \(f(ab)=f(a)f(b)\) (where \(a,b\) are arbitrary elements of \(G\)), and that is bijective2. When two groups are isomorphic, it means that their structures as groups are exactly identical. In other words, the multiplication tables of isomorphic groups coincide when symbols are appropriately substituted.
Point Groups (Finite Groups)
For example, comparing the multiplication tables of the point groups \(\bar{1}\) and \(m\), we can see they become equal under the substitution \(\bar{1} \leftrightarrow m_{010}\). That is, the point groups \(\bar{1}\) and \(m\) are isomorphic. The multiplication table of a group of order (number of elements) 2, with the identity element as \(E\) and the other element as \(A\), can only have the following pattern. The point groups of order 2 are \(\bar{1}, m, 2\), and they are all isomorphic.
| \(E\) | \(A\) | |
| \(E\) | \(E\) | \(A\) |
| \(A\) | \(A\) | \(E\) |
On the other hand, no matter how the symbols in the multiplication table of the point group \(\bar{4}\) are substituted, they cannot be made to coincide with the multiplication table of the point group \(222\). Therefore, the point groups \(\bar{4}\) and \(222\) are not isomorphic. The multiplication table of a group of order 4, with the identity element as \(E\) and the other elements as \(A, B, C\), has the following two patterns.
| \(E\) | \(A\) | \(B\) | \(C\) | |
| \(E\) | \(E\) | \(A\) | \(B\) | \(C\) |
| \(A\) | \(A\) | \(E\) | \(C\) | \(B\) |
| \(B\) | \(B\) | \(C\) | \(A\) | \(E\) |
| \(C\) | \(C\) | \(B\) | \(E\) | \(A\) |
| \(E\) | \(A\) | \(B\) | \(C\) | |
| \(E\) | \(E\) | \(A\) | \(B\) | \(C\) |
| \(A\) | \(A\) | \(E\) | \(C\) | \(B\) |
| \(B\) | \(B\) | \(C\) | \(E\) | \(A\) |
| \(C\) | \(C\) | \(B\) | \(A\) | \(E\) |
The former corresponds to the point groups \(4, \bar{4}\), and the latter corresponds to the point groups \(2/m, 222, mm2\). For a detailed classification of isomorphisms for groups of finite order (finite groups), see this resource (materials by Prof. Hanaki at Shinshu University).
Space Groups (Infinite Groups)
Now, we have established that the point groups \(\bar{1}\) and \(m\) are isomorphic. Then, are the space groups \(P\bar{1}\) and \(Pm\) isomorphic? Some might think “since we just added lattice translations, they should remain isomorphic,” but this is actually not the case. Let us write out and examine their multiplication tables (since space groups have infinite order, we will write only a portion).
Multiplication table of \(P\bar{1}\)
| \(\{1|0,0,0\}\) | \(\{\bar{1}|0,0,0\}\) | \(\{1|1,0,0\}\) | \(\{\bar{1}|1,0,0\}\) | \(\{1|0,1,0\}\) | \(\{\bar{1}|0,1,0\}\) | \(\cdots\) | |
| \(\{1|0,0,0\}\) | \(\{1|0,0,0\}\) | \(\{\bar{1}|0,0,0\}\) | \(\{1|1,0,0\}\) | \(\{\bar{1}|1,0,0\}\) | \(\{1|0,1,0\}\) | \(\{\bar{1}|0,1,0\}\) | \(\cdots\) |
| \(\{\bar{1}|0,0,0\}\) | \(\{\bar{1}|0,0,0\}\) | \(\{1|0,0,0\}\) | \(\{\bar{1}|1,0,0\}\) | \(\{1|1,0,0\}\) | \(\{\bar{1}|0,1,0\}\) | \(\{1|0,1,0\}\) | \(\cdots\) |
| \(\{1|1,0,0\}\) | \(\{1|1,0,0\}\) | \(\{\bar{1}|\bar{1},0,0\}\) | \(\{1|2,0,0\}\) | \(\{\bar{1}|0,0,0\}\) | \(\{1|1,1,0\}\) | \(\{\bar{1}|\bar{1},1,0\}\) | \(\cdots\) |
| \(\{\bar{1}|1,0,0\}\) | \(\{\bar{1}|1,0,0\}\) | \(\{1|\bar{1},0,0\}\) | \(\{\bar{1}|2,0,0\}\) | \(\{1|0,0,0\}\) | \(\{\bar{1}|1,1,0\}\) | \(\{1|\bar{1},1,0\}\) | \(\cdots\) |
| \(\{1|0,1,0\}\) | \(\{1|0,1,0\}\) | \(\{\bar{1}|0,\bar{1},0\}\) | \(\{1|1,1,0\}\) | \(\{\bar{1}|1,\bar{1},0\}\) | \(\{1|0,2,0\}\) | \(\{\bar{1}|0,0,0\}\) | \(\cdots\) |
| \(\{\bar{1}|0,1,0\}\) | \(\{\bar{1}|0,1,0\}\) | \(\{1|0,\bar{1},0\}\) | \(\{\bar{1}|1,1,0\}\) | \(\{1|1,\bar{1},0\}\) | \(\{\bar{1}|0,2,0\}\) | \(\{1|0,0,0\}\) | \(\cdots\) |
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\ddots\) |
Multiplication table of \(Pm\) (principal axis \(b\))
| \(\{1|0,0,0\}\) | \(\{m_{010}|0,0,0\}\) | \(\{1|1,0,0\}\) | \(\{m_{010}|1,0,0\}\) | \(\{1|0,1,0\}\) | \(\{m_{010}|0,1,0\}\) | \(\cdots\) | |
| \(\{1|0,0,0\}\) | \(\{1|0,0,0\}\) | \(\{m_{010}|0,0,0\}\) | \(\{1|1,0,0\}\) | \(\{m_{010}|1,0,0\}\) | \(\{1|0,1,0\}\) | \(\{m_{010}|0,1,0\}\) | \(\cdots\) |
| \(\{m_{010}|0,0,0\}\) | \(\{m_{010}|0,0,0\}\) | \(\{1|0,0,0\}\) | \(\{m_{010}|1,0,0\}\) | \(\{1|1,0,0\}\) | \(\{m_{010}|0,1,0\}\) | \(\{1|0,1,0\}\) | \(\cdots\) |
| \(\{1|1,0,0\}\) | \(\{1|1,0,0\}\) | \(\{m_{010}|1,0,0\}\) | \(\{1|2,0,0\}\) | \(\{m_{010}|2,0,0\}\) | \(\{1|1,1,0\}\) | \(\{m_{010}|1,1,0\}\) | \(\cdots\) |
| \(\{m_{010}|1,0,0\}\) | \(\{m_{010}|1,0,0\}\) | \(\{1|1,0,0\}\) | \(\{m_{010}|2,0,0\}\) | \(\{1|2,0,0\}\) | \(\{m_{010}|1,1,0\}\) | \(\{1|1,1,0\}\) | \(\cdots\) |
| \(\{1|0,1,0\}\) | \(\{1|0,1,0\}\) | \(\{m_{010}|0,\bar{1},0\}\) | \(\{1|1,1,0\}\) | \(\{m_{010}|1,\bar{1},0\}\) | \(\{1|0,2,0\}\) | \(\{m_{010}|0,0,0\}\) | \(\cdots\) |
| \(\{m_{010}|0,1,0\}\) | \(\{m_{010}|0,1,0\}\) | \(\{1|0,\bar{1},0\}\) | \(\{m_{010}|1,1,0\}\) | \(\{1|1,\bar{1},0\}\) | \(\{m_{010}|0,2,0\}\) | \(\{1|0,0,0\}\) | \(\cdots\) |
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\ddots\) |
The unfilled portions represent operations where the linear part is \(1\) (that is, simple parallel translation operations), and they naturally match completely. What is noteworthy is the portions with blue backgrounds. Let us focus on these portions and compare them carefully. Then, you should notice that they cannot be made equal through symbol substitution. This is because, for example, in the space group \(P\bar{1}\), squaring \(\{\bar{1}|u,v,w\}\) always yields \(\{1|0,0,0\}\), but in the space group \(Pm\), squaring \(\{m_{010}|u,v,w\}\) does not necessarily yield \(\{1|0,0,0\}\)3. Therefore, while the point groups \(\bar{1}\) and \(m\) are isomorphic, the space groups \(P\bar{1}\) and \(Pm\) are not isomorphic. The fact that adding translational operations can make isomorphic point groups non-isomorphic is a general property of groups.
Isomorphic Space Groups
Basically, space groups with different notations are in a non-isomorphic relationship, but there are exceptions. Among the 230 types of three-dimensional space groups, the following 11 pairs are mutually isomorphic.
- Tetragonal: \(P4_1\) and \(P4_3\), \(P4_122\) and \(P4_322\), \(P4_12_12\) and \(P4_32_12\)
- Trigonal: \(P3_1\) and \(P3_2\), \(P3_121\) and \(P3_221\), \(P3_112\) and \(P3_212\)
- Hexagonal: \(P6_1\) and \(P6_5\), \(P6_2\) and \(P6_4\), \(P6_122\) and \(P6_522\), \(P6_222\) and \(P6_422\)
- Cubic: \(P4_132\) and \(P4_332\)
Each pair has an enantiomorphic relationship. The 219 space groups obtained by excluding the isomorphic overlap of these 11 pairs are called affine space-group types. On the other hand, the 230 space groups that allow isomorphic overlap are called proper affine space-group types.