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Classification of Subgroups of Space Groups

In general, space groups change with phase transitions in crystals. When such phase transitions occur due to atomic ordering/disordering phenomena or slight displacements, the space groups before and after the transition have a subgroup/supergroup relationship. Here, we explain the relationships (classifications and properties) of subgroup/supergroup relationships between space groups¹. To clarify the hierarchical structure of space groups, we only consider maximal subgroups/minimal supergroups².

First, we classify subgroups of space groups into the following two categories.

1. Translation operations are preserved but some other symmetry operations (rotation, rotoinversion, screw, glide reflection) are lost, subgroups where the crystal class changes (t-subgroups)
2. Some translation operations are lost but other symmetry operations are preserved, subgroups where the crystal class does not change (k-subgroups)

Case 1 is called a t-subgroup, and case 2 is called a k-subgroup³. Of course, there are also subgroups where both translation operations and other operations are lost, but these can be considered as “k-subgroups of a t-subgroup,” so classification is unnecessary and they cannot be maximal subgroups. t-subgroups and k-subgroups are sometimes called Type I and Type II depending on the notation convention⁴. Furthermore, Type II is divided into three subtypes for practical reasons, but the definition differs slightly between International Tables for Crystallography Volume A (5^th edition, 2002, hereinafter ITA 5^th) and earlier editions and Volume A1 (1^st edition, 2004, hereinafter ITA1 1^st) and later editions. Furthermore, this subtype is also related to the concept of isomorphic subgroup (i-subgroup). Since the classification of subgroups of space groups is complicated in this way, I have summarized it in a table below.

The following supplementary explanations are provided for important terms.

Crystal Classes

As already described on the page “5.1. Classification of Space Groups“, in short, it refers to the crystallographic point groups. All space groups correspond to one of 32 types of crystallographic point groups, which are called crystal classes. This is a term that is easily confused with Crystal System, so please be careful. The crystal class is obtained from the space group notation by: ① deleting the first letter (lattice symbol), ② deleting all subscript numbers (translation information of screw axes), and ③ converting all lowercase letters (glide reflection operations) to \(m\). For example, space group \(Fd\bar{3}m\) is crystal class \(m\bar{3}m\), and space group \(P6_422\) is crystal class \(622\).

More precisely, the concept of crystal class refers to a point group with the same algebraic structure as the quotient group taken modulo the translation group, which is a subgroup of the space group. For detailed discussions on this, please see the page “2.4. Concepts of Subgroups and Quotient Groups“.

Isomorphic Subgroups

These are subgroups that have an isomorphic relationship. They are sometimes expressed as i-subgroups. Isomorphic means the relationship between groups \(G_1\) and \(G_2\) when there exists a mapping \(f\) that establishes a one-to-one correspondence between elements of \(G_1\) and elements of \(G_2\). The mapping \(f\) must be a bijective function. If groups \(G_1\) and \(G_2\) are isomorphic, they can be considered as groups with exactly the same properties from a group-theoretic perspective (for the concept of isomorphism, please also see the page “2.5. Group Multiplication Table (Cayley Table) and Isomorphism“).

Space groups with identical notation are necessarily isomorphic, and space groups with different notation are fundamentally not isomorphic⁵. Therefore, i-subgroups usually have the same notation as the original space group, but the basis vectors are necessarily different.

I (I-Maximal subgroup)

The definition is identical to “t-“, and it is expressed with the symbol “I” in both ITA 5^th and ITA1 1^st. It is a group from which only some point group operations are removed, and the basic translation vectors do not change (i.e., the unit cell is the same), but the crystal class changes. Because some point group operations are removed, it cannot have an isomorphic relationship.

For example, \(P1\) with respect to \(P2\), or \(F\bar{4}3m\) with respect to \(Fd\bar{3}m\), belong to this type.

IIa = II (Loss of centring translations)

This is a type of k-subgroup and refers to a subgroup obtained by removing the composite lattice component from the original space group. It is expressed as IIa in ITA 5^th, and as II (Loss of centring translations) in ITA1 1^st.

If the original space group is a \(P\) lattice, subgroups of this type do not exist. If it is any of \(I\), \(F\), \(A\), \(B\), \(C\), \(R\) (for the hexagonal setting), subgroups of this type necessarily exist. Since we only remove the composite lattice component, the unit cell size does not change. Also, it cannot have an isomorphic relationship.

For example, \(P2\) with respect to \(C2\), or \(Pm\bar{3}m\) with respect to \(Im\bar{3}m\), belong to this type.

II (Enlarged unit cell)

This is a type of k-subgroup. It refers to a subgroup obtained by removing some of the translation operations so that the unit cell becomes larger. It is the union of IIb and IIc described below. Subgroups of this type exist infinitely.

IIb

Among subgroups classified as II (Enlarged unit cell), these are subgroups that are not isomorphic.

IIc = II (Series of maximal isomorphic subgroups)

Among subgroups classified as II (Enlarged unit cell), these are subgroups that are isomorphic. It is expressed as IIc in ITA 5^th, and as II (Series of maximal isomorphic subgroups) in ITA1 1^st.

For example, with respect to \(P2\) (principal axis is \(b\)), \(P2\) and \(P2_1\) with crystal lattices with the \(b\)-axis doubled are both maximal subgroups of the II (Enlarged unit cell) type. \(P2\) is an isomorphic subgroup so it is IIc, and \(P2_1\) is a non-isomorphic subgroup so it is IIb.

Examples of Subgroups

The hierarchical structure of subgroups for the 230 space groups is documented on separate pages (t-subgroups, k-subgroups). Here, we explain the properties of maximal subgroups by giving specific examples of several space groups. The explanation follows the notation method in ITA 5^th.

Example 1: \(P1\)

I type

\(P1\) does not include any point group operations other than the identity operation, so there are no subgroups of this type.

IIa type

\(P1\) is not a composite lattice, so this type does not exist either.

IIb type

Since the only subgroup of \(P1\) can only be \(P1\), this type does not exist either.

IIc type

Subgroups of \(P1\) are only of this type. For example, with respect to space group \(P1\) with crystal lattice vectors \(\textbf{a},\textbf{b}, \textbf{c}\), space group \(P1\) with crystal lattice vectors \(p\textbf{a},q\textbf{b}, r\textbf{c}\) (where \(p,q,r\) are coprime natural numbers) is an IIc subgroup, and its index is \(p\times q\times r\). If \(p,q,r\) are not coprime, it becomes a non-maximal subgroup.

Also, if we set a new lattice by appropriately combining \(\textbf{a},\textbf{b}, \textbf{c}\), it again becomes an IIc subgroup. For example, \(\textbf{a},\textbf{b}+\textbf{c}, \textbf{c}-\textbf{b}\), or \(\textbf{a}-\textbf{c},\textbf{b}, \textbf{a}+\textbf{c}\), or \(\textbf{a}+\textbf{b},\textbf{b}-\textbf{a}, \textbf{c}\), or \(\textbf{b}+\textbf{c},\textbf{c}+\textbf{a}, \textbf{a}+\textbf{b}\), etc. (all with index 2).

Example 2: \(C1m1\)

The general positions⁶ in one unit cell of \(C1m1\) are the following four.

\((1) x, y, z \,\, (2)x,\bar{y},z \,\, (3)x+\frac{1}{2},y+\frac{1}{2},z \,\, (4)x+\frac{1}{2},\bar{y}+\frac{1}{2},z\)

The operation connecting \((1)\) and \((3)\) (or \((2)\) and \((4)\)) is \(C\), namely \((\frac{1}{2},\frac{1}{2},0)+ \). The operation connecting \((1)\) and \((2)\) (or \((3)\) and \((4)\)) is the \(m\) mirror reflection, and the operation connecting \((1)\) and \((4)\) (or \((2)\) and \((3)\)) is the \(a\) glide reflection.

I type

I type is a subgroup where translation operations are preserved but some point group operations are lost. A subgroup created by preserving translation operations (that is, keeping \((1)\) and \((3)\)) is \(C1\)⁷, and this is the only I type. The point group operation \(m\) is indeed lost.

IIa type

IIa type is a subgroup where some translation operations are lost but the unit cell size does not change. The composite lattice is \(C\), so at first glance \(P1m1\) seems to be the only answer, but actually there is another one: \(P1a1\). If we keep \((1)\) and \((2)\), we get \(P1m1\), and if we keep \((1)\) and \((4)\), we get \(P1a1\). Both lose the translation operation \(C : (\frac{1}{2},\frac{1}{2},0)+ \), so they are subgroups of IIa.

IIb, IIc type

IIb and IIc are both subgroups where some translation operations are lost from the original group and the unit cell becomes larger. The case where it is not isomorphic to the original group is IIb, and the case where it is isomorphic is IIc. For \(C1m1\), we can set new lattices by multiplying or combining the crystal lattice vectors \(\textbf{a},\textbf{b}, \textbf{c}\), and then remove translation operations from there. For example, let’s consider a crystal lattice where the \(\textbf{c}\) direction is doubled. The general positions in the unit cell with twice the original size are

$$\begin{array}{lll}
(1) x, y, z & (2)x,\bar{y},z &(3)x+\frac{1}{2},y+\frac{1}{2},z & (4)x+\frac{1}{2},\bar{y}+\frac{1}{2},z\ \\
(5) x, y, z+\frac{1}{2} & (6)x,\bar{y},z+\frac{1}{2} & (7)x+\frac{1}{2},y+\frac{1}{2},z+\frac{1}{2} & (8)x+\frac{1}{2},\bar{y}+\frac{1}{2},z+\frac{1}{2}
\end{array}$$

can be expressed. Since we reset \(2\textbf{c}\) to \(\textbf{c}\), note that \((5)\sim(8)\) have \(+\frac{1}{2}\) added to the Z coordinate relative to \((1)\sim(4)\). If we keep \((1), (2), (3), (4)\) from these, we can extract an isomorphic subgroup \(C1m1\) (IIc), and if we keep \((1), (3), (6), (8)\), we can extract a non-isomorphic subgroup \(C1c1\) (IIb).

There are still many IIb and IIc subgroups of \(C1m1\), but it is very tedious to comprehensively cover the process of multiplying and combining \(\textbf{a},\textbf{b}, \textbf{c}\). Unless you are very enthusiastic about group theory, it is best to refer to the information summarized in ITA.

Footnotes

1. Of course, it is also possible to extract point groups as subgroups from space groups. Such point groups are called site symmetry. For details, please refer to “3.3. Site Symmetry and Wyckoff Positions“. ↩︎
2. These are sometimes translated as “greatest subgroup / smallest supergroup,” but on this website we use the terms maximal subgroup / minimal supergroup. The requirement for a subgroup \(H\) of a group \(G\) to be a maximal subgroup is that no subgroup of \(G\) that is a supergroup of \(H\) exists (excluding \(G\) and \(H\) themselves). ↩︎
3. “t-” is the initial letter of the German word translationengleich (meaning “same translations”), and “k-” is the initial letter of klassengleiche (meaning “same class”). ↩︎
4. Crystallographers in English-speaking countries seem to prefer the Type I, II notation. ITA also adopts the Type I, II notation. ↩︎
5. The exceptions are: (1) cases where the space group notation appears to change due to a transformation of basis vectors (changing only the choice of \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) without changing the crystal lattice) (e.g., \(Pma2\) and \(P2mb\), see “6.3. Choice of Axes, Transformation of Axes“), and (2) cases of enantiomorphic space groups (e.g., \(P4_1\) and \(P4_3\), see “2.4. Multiplication Tables and Isomorphism of Groups“). ↩︎
6. A general position is a Wyckoff position whose site symmetry is \(1\). For the concept of Wyckoff positions, please refer to “3.3. Site Symmetry and Wyckoff Positions“. ↩︎
7. Incidentally, \(C1\) is not a standard space group notation. By transforming the crystal lattice vectors \(\textbf{a}, \textbf{b}, \textbf{c}\) of \(C1\) to \(\frac{1}{2}(\textbf{a}-\textbf{b}), \frac{1}{2}(\textbf{a}+\textbf{b}), \textbf{c}\), it can be expressed as the standard notation space group \(P1\). ↩︎