International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 4-5

Section 1.1.3. Subgroups

B. Souvigniera*

aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

1.1.3. Subgroups

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The group of symmetry operations of a crystal pattern may alter if the crystal undergoes a phase transition. Often, some symmetries are preserved, while others are lost, i.e. symmetry breaking takes place. The symmetry operations that are preserved form a subset of the original symmetry group which is itself a group. This gives rise to the concept of a subgroup.

Definition. A subset [{\cal H} \subseteq {\cal G}] is called a subgroup of [{\cal G}] if its elements form a group by themselves. This is denoted by [{\cal H} \leq {\cal G}]. If [{\cal H}] is a subgroup of [{\cal G}], then [{\cal G}] is called a supergroup of [{\cal H}]. In order to be a subgroup, [{\cal H}] is required to contain the identity element [\ispecialfonts{\sfi e}] of [{\cal G}], to contain inverse elements and to be closed with respect to composition of elements. Thus, technically, every group is a subgroup of itself.

The subgroups of [{\cal G}] that are not equal to [{\cal G}] are called proper subgroups of [{\cal G}]. A proper subgroup [{\cal H}] of [{\cal G}] is called a maximal subgroup if it is not a proper subgroup of any proper subgroup [{\cal H}'] of [{\cal G}].

It is often convenient to specify a subgroup [{\cal H}] of [{\cal G}] by a set [\ispecialfonts\{ {\sfi h}_1, \ldots, {\sfi h}_s \}] of generators. This is denoted by [\ispecialfonts{\cal H} = \langle {\sfi h}_1, \ldots, {\sfi h}_s \rangle]. The order of [{\cal H}] is not a priori obvious from the set of generators. For example, in the symmetry group 4mm of the square the pairs [\{ m_{10}, m_{01} \}] and [\{m_{11}, m_{1\bar1} \}] both generate subgroups of order 4, whereas the pair [\{ m_{10}, m_{11} \}] generates the full group of order 8.

The subgroups of a group can be visualized in a subgroup diagram. In such a diagram the subgroups are arranged with subgroups of higher order above subgroups of lower order. Two subgroups are connected by a line if one is a maximal subgroup of the other. By following downward paths in this diagram, all group–subgroup relations in a group can be derived. Additional information is provided by connecting subgroups of the same order by a horizontal line if they are conjugate (see Section 1.1.7[link]).

Examples

  • (i) The set [\ispecialfonts\{ {\sfi e} \}] consisting only of the identity element of [{\cal G}] is a subgroup, called the trivial subgroup of [{\cal G}].

  • (ii) For the group [\bb Z] of the integers, all subgroups are cyclic and generated by some integer n, i.e. they are of the form [n {\bb Z}: = \{ n a \mid a \in {\bb Z} \}] for an integer n. Such a subgroup is maximal if n is a prime number.

  • (iii) For every element [\ispecialfonts{\sfi g}] of a group [{\cal G}], the powers of [\ispecialfonts{\sfi g}] form a subgroup of [{\cal G}] which is a cyclic group.

  • (iv) In [{\rm GL}_n({\bb R})] the matrices of determinant 1 form a subgroup, since the determinant of the matrix product [{\bi A} \cdot {\bi B}] is equal to the product of the determinants of [{\bi A}] and [{\bi B}].

  • (v) In the symmetry group 3m of an equilateral triangle the rotations form a subgroup of order 3 (see Fig. 1.1.3.1[link]).

    [Figure 1.1.3.1]

    Figure 1.1.3.1 | top | pdf |

    Subgroup diagram for the symmetry group 3m of an equilateral triangle.

  • (vi) The symmetry group 2mm of a rectangle has three subgroups of order 2, generated by the reflection m10, the twofold rotation 2 and the reflection m01, respectively (see Fig. 1.1.3.2[link]).

    [Figure 1.1.3.2]

    Figure 1.1.3.2 | top | pdf |

    Subgroup diagram for the symmetry group 2mm of a rectangle.

  • (vii) In the symmetry group 4mm of the square, the reflections m10 and m01 together with their product 2 and the identity element 1 form a subgroup of order 4. This subgroup can be recognized in the subgroup diagram of 4mm as the subdiagram of the subgroups of [\langle 2, {m}_{10}\rangle] in the left part of Fig. 1.1.3.3[link] which coincides with the subgroup diagram of 2mm in Fig. 1.1.3.2[link]. A different subgroup of order 4 is formed by the other pair of perpendicular reflections [m_{11}], [m_{1\bar1}] together with 2 and 1 and a third subgroup of order 4 is the cyclic subgroup [\langle {4}^+\rangle] generated by the fourfold rotation (see Fig. 1.1.3.3[link]).

    [Figure 1.1.3.3]

    Figure 1.1.3.3 | top | pdf |

    Subgroup diagram for the symmetry group 4mm of the square.








































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