Series of maximal isomorphic subgroups

Billiet, Y.

doi:10.1107/97809553602060000542

2.1.5.1. General description

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Maximal subgroups of index higher than 4 have index p, [p^2] or [p^3] , where p is prime, are necessarily isomorphic subgroups and are infinite in number. Only a few of them are listed in IT A in the block `Maximal isomorphic subgroups of lowest index IIc'. Because of their infinite number, they cannot be listed individually, but are listed in this volume as members of series under the heading `Series of maximal isomorphic subgroups'. In most of the series, the HM symbol for each isomorphic subgroup $[{\cal H} \,\lt\, {\cal G}]$ will be the same as that of $[{\cal G}]$ . However, if $[{\cal G}]$ is an enantiomorphic space group, the HM symbol of $[{\cal H}]$ will be either that of $[{\cal G}]$ or that of its enantiomorphic partner.

Example 2.1.5.1.1

Two of the four series of isomorphic subgroups of the space group [P4_1] , No. 76, are (the data on the generators are omitted):

[p]	$[{\bf c}'=p{\bf c}]$
	$[P4_3\,\,(78)]$	; $[p\equiv 3]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups
	$[P4_1\,\,(76)]$	; $[p\equiv 1]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups

On the other hand, the corresponding data for [P4_3] , No. 78, are

[p]	$[{\bf c}'=p{\bf c}]$
	$[P4_3\,\,(78)]$	; $[p\equiv 1]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups
	$[P4_1\,\,(76)]$	; $[p\equiv 3]$ (mod 4)	$[{\bf a},{\bf b},p{\bf c}]$
		no conjugate subgroups

Note that in both tables the subgroups of the type [P4_3] , No. 78, are listed first because of the rules on the sequence of the subgroups.

If an isomorphic maximal subgroup of index $[i \leq 4 ]$ is a member of a series, then it is listed twice: as a member of its series and individually under the heading `Enlarged unit cell'.

Most isomorphic subgroups of index 3 are the first members of series but those of index 2 or 4 are rarely so. An example is the space group [P4_2] , No. 77, with isomorphic subgroups of index 2 (not in any series) and 3 (in a series); an exception is found in space group [P4] , No. 75, where the isomorphic subgroup for $[{\bf c}'=2{\bf c}]$ is the first member of the series $[[p]\, {\bf c}'=p{\bf c}]$ .

2.1.5.2. Basis transformation

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The conventional basis of the unit cell of each isomorphic subgroup in the series has to be defined relative to the basis of the original space group. For this definition the prime p is frequently sufficient as a parameter.

Example 2.1.5.2.1

The isomorphic subgroups of the space group [P4_222] , No. 93, can be described by two series with the bases of their members:

$[[p]\quad{\bf a}, \,{\bf b}, \, p{\bf c}]$
$[[p^2]\quad p{\bf a},\ p{\bf b},\ {\bf c}]$ .

In other cases, one or two positive integers, here called q and r, define the series and often the value of the prime p.

Example 2.1.5.2.2

In space group $[P\overline{6}]$ , No. 174, the series $[q{\bf a}-r{\bf b},\, r{\bf a}+(q+r){\bf b},\, {\bf c}]$ is listed. The values of q and r have to be chosen such that while $[q>0,\ r>0,\ p=q^2+r^2+qr]$ and p is prime.

Example 2.1.5.2.3

In the space group [P112_1/m] , No. 11, unique axis c, the series $[p{\bf a},\, -q{\bf a}+{\bf b},\, {\bf c}]$ is listed. Here p and q are independent and q may take the p values $[0\leq q \,\lt\, p]$ for each value of p.

2.1.5.3. Origin shift

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Each of the sublattices discussed in Section 2.1.4.3.2 is common to a conjugacy class or belongs to a normal subgroup of a given series. The subgroups in a conjugacy class differ by the positions of their conventional origins relative to the origin of the space group $[{\cal G}]$ . To define the origin of the conventional unit cell of each subgroup in a conjugacy class, one, two or three integers, called u, v or w in these tables, are necessary. For a series of subgroups of index p, [p^2] or [p^3] there are p, [p^2] or [p^3] conjugate subgroups , respectively. The positions of their origins are defined by the p or [p^2] or [p^3] permitted values of u or u, v or u, v, w, respectively.

Example 2.1.5.3.1

The space group $[{\cal G}]$ , $[P\overline{4}2c]$ , No. 112, has two series of maximal isomorphic subgroups $[{\cal H}]$ . For one of them the lattice relations are $[[p^2]\, {\bf a}'=p{\bf a},{\bf b}'=p{\bf b}]$ , listed as $[p{\bf a},p{\bf b},{\bf c}]$ for the transformation matrix. The index is [p^2] . For each value of p there exist exactly [p^2] conjugate subgroups with origins in the points $[u,\,v,\,0]$ , where the parameters u and v run independently: $[0\leq u \,\lt\, p]$ and $[0\leq v \,\lt\, p]$ .

In another type of series there is exactly one (normal) subgroup $[{\cal H}]$ for each index p; the location of its origin is always chosen at the origin $[0,\,0,\,0]$ of $[{\cal G}]$ and is thus not indicated as an origin shift.

Example 2.1.5.3.2

Consider the space group [Pca2_1] , No. 29. Only one subgroup exists for each value of p in the series $[{\bf a}, {\bf b},p{\bf c}]$ . This is indicated in the tables by the statement `no conjugate subgroups'.

2.1.5.4. Generators

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The generators of the p (or [p^2] or [p^3] ) conjugate isomorphic subgroups $[{\cal H}]$ are obtained from those of $[{\cal G}]$ by adding translational components. These components are determined by the parameters p (or q and r, if relevant) and u (and v and w, if relevant).

Example 2.1.5.4.1

Space group [P2_13] , No. 198.

In the series defined by the lattice relations $[p{\bf a},p{\bf b}, p{\bf c}]$ and the origin shift $[u,\,v,\,w]$ there exist exactly [p^3] conjugate subgroups for each value of p. The generators of each subgroup are defined by the parameter p and the triplet $[u,\,v,\,w]$ in combination with the generators (2), (3) and (5) of $[{\cal G}]$ . Consider the subgroup characterized by the basis $[7{\bf a}, 7{\bf b},7{\bf c}]$ and by the origin shift $[u=3,\ v=4,\ w=6]$ . One obtains from the generator (2) $[\overline{x}+{{1}\over{2}},\overline{y},z+{{1}\over{2}}]$ of $[{\cal G}]$ the corresponding generator of $[{\cal H}]$ by adding the translation vector $[({{p}\over{2}}-{{1}\over{2}}+2u){\bf a}+2v{\bf b}+({{p}\over{2}}- {{1}\over{2}}){\bf c}]$ to the translation vector $[{{1}\over{2}}{\bf a}+ {{1}\over{2}}{\bf c}]$ of the generator (2) of $[{\cal G}]$ and obtains $[{{19}\over{2}}\,{\bf a}+8{\bf b}+{{7}\over{2}}\,{\bf c}]$ , so that this generator of $[{\cal H}]$ is written $[\overline{x}+{{19}\over{2}},\overline{y}+8,z+{{7}\over{2}}]$ .

2.1.5.5. Special series

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For most space groups, there is only one description of their series of the isomorphic subgroups. However, if a space group is described twice in IT A, then there are also two different descriptions of these series. This happens for monoclinic space groups with the settings unique axis b and unique axis c, for some orthorhombic, tetragonal and cubic space groups with origin choice 1 and origin choice 2 and for trigonal space groups with rhombohedral lattices with hexagonal axes and rhombohedral axes.

2.1.5.5.1. Monoclinic space groups

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In the monoclinic space groups, the series in the listings `unique axis b' and `unique axis c' are closely related by a simple cyclic permutation of the axes a, b and c, see IT A, Section 2.2.16 .

2.1.5.5.2. Trigonal space groups with rhombohedral lattice

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In trigonal space groups with rhombohedral lattices, the series with hexagonal axes and with rhombohedral axes appear to be rather different. However, the `rhombohedral' series are the exact transcript of the `hexagonal' series by the same transformation formulae as are used for the different monoclinic settings. However, the transformation matrices P and $[\bi{P}^{-1}]$ in Part 5 of IT A are more complicated in this case.

Example 2.1.5.5.1

Space group $[R\overline{3}]$ , No. 148. The second series is described with hexagonal axes by the basis transformation $[{\bf a},\, {\bf b},\, p{\bf c}]$ , i.e. $[{\bf a}'_{\rm hex}={\bf a}_{\rm hex},\, {\bf b}'_{\rm hex}={\bf b}_{\rm hex},\, {\bf c}'_{\rm hex}=p\,{\bf c}_{\rm hex},]$ and the origin shift [0,0,u] . We discuss the basis transformation first. It can be written $[ ({\bf a}_{\rm hex}')^{\rm T}=({\bf a}_{\rm hex})^{\rm T}\bi{X} \eqno (2.1.5.1)]$ in analogy to Part 5 , IT A. Here $[({\bf a}_{\rm hex})^{\rm T}]$ is the row of basis vectors of the conventional hexagonal basis. The matrix X is defined by $[{\bi X}=\left(\matrix { 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & p } \right).]$ With rhombohedral axes, equation (2.1.5.1) would be written $[({\bf a}_{\rm rh}')^{\rm T}=({\bf a}_{\rm rh})^{\rm T}{\bi Y}, \eqno (2.1.5.2)]$ with the matrix Y to be determined.

The transformation from hexagonal to rhombohedral axes is described by $[({\bf a}_{\rm rh})^{\rm T}=({\bf a}_{\rm hex})^{\rm T}{\bi P}^{-1}, \eqno (2.1.5.3)]$ where the matrices $[{\bi P}^{-1}=\left(\matrix { {{2}\over{3}} & -{{1}\over{3}} & -{{1}\over{3}} \cr {{1}\over{3}} & {{1}\over{3}} & -{{2}\over{3}} \cr {{1}\over{3}} & {{1}\over{3}} & {{1}\over{3}} } \right) \,\, {\rm and}\, \,{\bi P}=\left(\matrix { 1 & 0 & 1 \cr \overline{1} & 1 & 1 \cr 0 & \overline{1} & 1 } \right)]$ are listed in IT A, Table 5.1.3.1 , see also Figs. 5.1.3.6 (a) and (c) in IT A.

Applying equations (2.1.5.3), (2.1.5.1) and (2.1.5.2), one gets $[({\bf a}'_{\rm rh})^{\rm T}=({\bf a}'_{\rm hex})^{\rm T}{\bi P}^{-1}= ({\bf a}_{\rm hex})^{\rm T} {\bi X}{\bi P}^{-1} =({\bf a}_{\rm rh})^{\rm T} {\bi Y}= ({\bf a}_{\rm hex})^{\rm T} {\bi P}^{-1}{\bi Y}. \eqno (2.1.5.4)]$ From equation (2.1.5.4) it follows that $[{\bi X}{\bi P}^{-1}={\bi P}^{-1}{\bi Y}\,\, {\rm or }\,\, {\bi Y}={\bi P}{\bi X}{\bi P}^{-1}. \eqno (2.1.5.5)]$ One obtains Y from equation (2.1.5.5) by matrix multiplication, $[{\bi Y}= \left(\matrix { {{p+2}\over{3}} & {{p-1}\over{3}} & {{p-1}\over{3}} \cr {{p-1}\over{3}} & {{p+2}\over{3}} & {{p-1}\over{3}} \cr {{p-1}\over{3}} & {{p-1}\over{3}} & {{p+2}\over{3}} } \right),]$ and from Y for the bases of the subgroups with rhombohedral axes $[\eqalign{{\bf a}_{\rm rh}' &=\textstyle{{1}\over{3}}[(p+2){\bf a}_{\rm rh}+(p-1){\bf b}_{\rm rh} +(p-1){\bf c}_{\rm rh}],\cr {\bf b}_{\rm rh}'&=\textstyle{{1}\over{3}}[(p-1){\bf a}_{\rm rh}+(p+2){\bf b}_{\rm rh} + (p-1){\bf c}_{\rm rh}],\cr {\bf c}_{\rm rh}'&=\textstyle{{1}\over{3}}[(p-1){\bf a}_{\rm rh}+(p-1){\bf b}_{\rm rh} +(p+2){\bf c}_{\rm rh}].}]$ The column of the origin shift $[{\bi u}_{\rm hex}=0,0,u]$ in hexagonal axes must be transformed by $[{\bi u}_{\rm rh}={\bi P}\,{\bi u}_{\rm hex}]$ . The result is the column $[{\bi u}_{\rm rh}=u,\,u,\,u]$ in rhombohedral axes.

2.1.5.5.3. Space groups with two origin choices

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Space groups with two origin choices are always described in the same basis, but origin 1 is shifted relative to origin 2 by the shift vector s. For most space groups with two origins, the appearance of the two series related by the origin shift is similar; there are only differences in the generators.

Example 2.1.5.5.2

Consider the space group [Pnnn] , No. 48, in both origin choices and the corresponding series defined by $[p{\bf a},\, {\bf b},\, {\bf c}]$ and $[u,\,0,\,0]$ . In origin choice 1, the generator (5) of $[{\cal G}]$ is described by the `coordinates' $[\overline{x}+{{1}\over{2}},\,\overline{y}+{{1}\over{2}},\,\overline{z}+{{1}\over{2}}]$ . The translation part $[({{p}\over{2}}-{{1}\over{2}}){\bf a}]$ of the third generator of $[{\cal H}]$ stems from the term $[{{1}\over{2}}]$ in the first `coordinate' of the generator (5) of $[{\cal G}]$ . Because $[({{p}\over{2}}-{{1}\over{2}}){\bf a}]$ must be a translation vector of $[{\cal G}]$ , p is odd. Such a translation part is not found in the generators (2) and (3) of $[{\cal H}]$ because the term $[{{1}\over{2}}]$ does not appear in the `coordinates' of the corresponding generators of $[{\cal G}]$ .

The situation is inverted in the description for origin choice 2.

The translation term $[({{p}\over{2}}-{{1}\over{2}}){\bf a}]$ appears in the first and second generator of $[{\cal H}]$ and not in the third one because the term $[{{1}\over{2}}]$ occurs in the first `coordinate' of the generators (2) and (3) of $[{\cal G}]$ but not in the generator (5).

The term [2u] appears in both descriptions. It is introduced in order to adapt the generators to the origin shift $[u,\,0,\,0]$ .

In other space groups described in two origin choices, surprisingly, the number of series is different for origin choice 1 and origin choice 2.

Example 2.1.5.5.3

In the tetragonal space group $[I4_{1}/amd]$ , No. 141, for origin choice 1 there is one series of maximal isomorphic subgroups of index [p^2] , p prime, with the bases $[p{\bf a},\, p{\bf b},\, {\bf c}]$ and origin shifts $[u,\,v,\,0]$ . For origin choice 2, there are two series with the same bases $[p{\bf a},\, p{\bf b},\, {\bf c}]$ but with the different origin shifts $[u,\,v,\,0]$ and $[{{1}\over{2}}+u,\,v,\,0]$ . What are the reasons for these results?

For origin choice 1, the term $[{{1}\over{2}}]$ appears in the first and second `coordinates' of all generators (2), (3), (5) and (9) of $[{\cal G}]$ . This term $[{{1}\over{2}}]$ is the cause of the translation vectors $[({{p}\over{2}}-{{1}\over{2}}){\bf a}]$ and $[({{p}\over{2}}-{{1}\over{2}}){\bf b}]$ in the generators of $[{\cal H}]$ .

For origin choice 2, fractions $[{{1}\over{4}}]$ and $[{{3}\over{4}}]$ appear in all `coordinates' of the generator (3) $[\overline{y}+{{1}\over{4}},x+{{3}\over{4}},z+{{1}\over{4}}]$ of $[{\cal G}]$ . As a consequence, translational parts with vectors $[({{p}\over{4}}+{{1}\over{4}}){\bf a}]$ and $[({{3p}\over{4}}- {{5}\over{4}}){\bf b}]$ appear if $[p\equiv3\hbox{ (mod 4)}]$ . On the other hand, translational parts with vectors $[({{p}\over{4}}-{{1}\over{4}}){\bf a},\ ({{3p}\over{4}}-{{3}\over{4}}){\bf b}]$ are introduced in the generators of $[{\cal H}]$ if $[p\equiv 1\hbox{ (mod 4)}]$ holds.

Another consequence of the fractions $[{{1}\over{4}}]$ and $[{{3}\over{4}}]$ occurring in the generator (3) of $[{\cal G}]$ is the difference in the origin shifts. They are $[{{1}\over{2}}+u,\,v,\,0]$ for $[p\equiv3\hbox{ (mod 4)}]$ and $[u,\,v,\,0]$ for $[p\equiv1]$ (mod 4). Thus, the one series in origin choice 1 for odd p is split into two series in origin choice 2 for $[p\equiv 3\hbox{ (mod 4)}]$ and $[p\equiv 1\hbox{ (mod 4)}]$ . ³

Section 2.1.5. Series of maximal isomorphic subgroups

2.1.5. Series of maximal isomorphic subgroups

2.1.5.1. General description

Example 2.1.5.1.1

2.1.5.2. Basis transformation

Example 2.1.5.2.1

Example 2.1.5.2.2

Example 2.1.5.2.3

2.1.5.3. Origin shift

Example 2.1.5.3.1

Example 2.1.5.3.2

2.1.5.4. Generators

Example 2.1.5.4.1

2.1.5.5. Special series

2.1.5.5.1. Monoclinic space groups

2.1.5.5.2. Trigonal space groups with rhombohedral lattice

Example 2.1.5.5.1

2.1.5.5.3. Space groups with two origin choices

Example 2.1.5.5.2

Example 2.1.5.5.3

References