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. 2-4

Section 1.1.2. Basic properties of groups

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

1.1.2. Basic properties of groups

| top | pdf |

Although groups occur in innumerable contexts, their basic properties are very simple and are captured by the following definition.

Definition. Let [{\cal G}] be a set of elements on which a binary operation is defined which assigns to each pair [\ispecialfonts({\sfi g},{\sfi h})] of elements the composition [\ispecialfonts{\sfi g} \circ {\sfi h} \in {\cal G}]. Then [{\cal G}], together with the binary operation [\circ], is called a group if the following hold:

  • (i) the binary operation is associative, i.e. [\ispecialfonts({\sfi g} \circ {\sfi h}) \circ {\sfi k} =] [\ispecialfonts {\sfi g} \circ ({\sfi h} \circ {\sfi k})];

  • (ii) there exists a unit element or identity element [\ispecialfonts{\sfi e} \in {\cal G}] such that [\ispecialfonts{\sfi g} \circ {\sfi e} = {\sfi g}] and [\ispecialfonts{\sfi e} \circ {\sfi g} = {\sfi g}] for all [\ispecialfonts{\sfi g} \in {\cal G}];

  • (iii) every [\ispecialfonts{\sfi g} \in {\cal G}] has an inverse element, denoted by [\ispecialfonts{\sfi g}^{-1}], for which [\ispecialfonts{\sfi g} \circ {\sfi g}^{-1} = {\sfi g}^{-1} \circ {\sfi g} = {\sfi e}].

In most cases, the composition of group elements is regarded as a product and is written as [\ispecialfonts{\sfi g} \cdot {\sfi h}] or even [\ispecialfonts{\sfi g} {\sfi h}] instead of [\ispecialfonts{\sfi g} \circ {\sfi h}]. An exception is groups where the composition is addition, e.g. a group of translations. In such a case, the composition [{\bf a} \circ {\bf b}] is more conveniently written as [{\bf a} + {\bf b}].

Examples

  • (i) The group consisting only of the identity element [\ispecialfonts{\sfi e}] (with [\ispecialfonts{\sfi e} \circ {\sfi e} = {\sfi e}]) is called the trivial group.

  • (ii) The group 3m of all symmetries of an equilateral triangle is a group with the composition of symmetry operations as binary operation. The group contains six elements, namely three reflections, two rotations and the identity element. It is schematically displayed in Fig. 1.1.2.2.

  • (iii) The set [\bb Z] of all integers forms a group with addition as operation. The identity element is 0, the inverse element for [a \in {\bb Z}] is [-a].

  • (iv) The set of complex numbers with absolute value 1 forms a circle in the complex plane, the unit circle [S^1]. The unit circle can be described by [S^1 = \{ \exp({2\pi i \, t}) \mid 0 \leq t \,\lt\, 1 \}] and forms a group with (complex) multiplication as operation.

  • (v) The set of all real [n \times n] matrices with determinant [\neq 0] is a group with matrix multiplication as operation. This group is called the general linear group and denoted by [{\rm GL}_n({\bb R})].

If a group [{\cal G}] contains finitely many elements, it is called a finite group and the number of its elements is called the order of the group, denoted by [|{\cal G}|]. A group with infinitely many elements is called an infinite group.

For a group element [\ispecialfonts{\sfi g}], its order is the smallest integer [n\,\gt\,0] such that [\ispecialfonts{\sfi g}^n = {\sfi e}] is the identity element. If there is no such integer, then [\ispecialfonts{\sfi g}] is said to be of infinite order.

The group operation is not required to be commutative, i.e. in general one will have [\ispecialfonts{\sfi g} {\sfi h} \neq {\sfi h} {\sfi g}]. However, a group [{\cal G}] in which [\ispecialfonts{\sfi g} {\sfi h} = {\sfi h} {\sfi g}] for all [\ispecialfonts{\sfi g}, {\sfi h}] is said to be a commutative or abelian group.

The inverse of the product [\ispecialfonts{\sfi g} {\sfi h}] of two group elements is the product of the inverses of the two elements in reversed order, i.e. [\ispecialfonts({\sfi g} {\sfi h})^{-1} = {\sfi h}^{-1} {\sfi g}^{-1}].

A particularly simple type of groups is cyclic groups in which all elements are powers of a single element [\ispecialfonts{\sfi g}]. A finite cyclic group [{\cal C}_n] of order n can be written as [{\cal C}_n =] [\ispecialfonts \{ {\sfi g}, {\sfi g}^2, \ldots, {\sfi g}^{n-1}, {\sfi g}^n = {\sfi e} \}]. For example, the rotations that are symmetry operations of an equilateral triangle constitute a cyclic group of order 3.

The group [\bb Z] of integers (with addition as operation) is an example of an infinite cyclic group in which negative powers also have to be considered, i.e. where [{\cal G} =] [\ispecialfonts \{ \ldots, {\sfi g}^{-2}, {\sfi g}^{-1}, {\sfi e} = {\sfi g}^0, {\sfi g}^1, {\sfi g}^2, \ldots \}].

Groups of small order may be displayed by their multiplication table, which is a square table with rows and columns indexed by the group elements and where the intersection of the row labelled by [\ispecialfonts{\sfi g}] and of the column labelled by [\ispecialfonts{\sfi h}] is the product [\ispecialfonts{\sfi g} {\sfi h}]. It follows immediately from the invertibility of the group elements that each row and column of the multiplication table contains every group element precisely once.

Examples

  • (i) A cyclic group of order 3 consists of the elements [\ispecialfonts\{ {\sfi g}, {\sfi g}^2, {\sfi g}^3 = {\sfi e} \}]. Its multiplication table is

      [\ispecialfonts\sfi e] [\ispecialfonts{\sfi g}] [\ispecialfonts{\sfi g}^2]
    [\ispecialfonts{\sfi e}] [\ispecialfonts{\sfi e}] [\ispecialfonts{\sfi g}] [\ispecialfonts{\sfi g}^2]
    [\ispecialfonts{\sfi g}] [\ispecialfonts{\sfi g}] [\ispecialfonts{\sfi g}^2] [\ispecialfonts{\sfi e}]
    [\ispecialfonts{\sfi g}^2] [\ispecialfonts{\sfi g}^2] [\ispecialfonts{\sfi e}] [\ispecialfonts{\sfi g}]

  • (ii) The symmetry group 2mm of a rectangle (with unequal sides) consists of a twofold rotation 2, two reflections [{m}_{10}, {m}_{01}] with mirror lines along the coordinate axes and the identity element 1 (see Fig. 1.1.2.1[link]; the small black lenticular symbol in the centre represents the twofold rotation point).

    [Figure 1.1.2.1]

    Figure 1.1.2.1 | top | pdf |

    Symmetry group 2mm of a rectangle.

    Note that in this and all subsequent examples of crystallographic point groups we will use the Seitz symbols (cf. Section 1.4.2.2[link] ) for the symmetry operations and the Hermann–Mauguin symbols (cf. Section 1.4.1[link] ) for the point groups.

    The multiplication table of the group 2mm is

      1 2 m10 m01
    1 1 2 m10 m01
    2 2 1 m01 m10
    m10 m10 m01 1 2
    m01 m01 m10 2 1

    The symmetry of the multiplication table (with respect to the main diagonal) shows that this is an abelian group.

  • (iii) The symmetry group 3m of an equilateral triangle consists (apart from the identity element 1) of the threefold rotations 3+ and 3 and the reflections m10, m01, m11 with mirror lines through a corner of the triangle and the centre of the opposite side (see Fig. 1.1.2.2[link]; the small black triangle in the centre represents the threefold rotation point).

    [Figure 1.1.2.2]

    Figure 1.1.2.2 | top | pdf |

    Symmetry group 3m of an equilateral triangle.

    The multiplication table of the group 3m is

      1 3+ 3 m10 m01 m11
    1 1 3+ 3 m10 m01 m11
    3+ 3+ 3 1 m11 m10 m01
    3 3 1 3+ m01 m11 m10
    m10 m10 m01 m11 1 3+ 3
    m01 m01 m11 m10 3 1 3+
    m11 m11 m10 m01 3+ 3 1

    The fact that [3^+ \cdot m_{10} = m_{11}], but [m_{10} \cdot 3^+ = m_{01}] shows that this group is not abelian. It is actually the smallest group (in terms of order) that is not abelian.

  • (iv) The symmetry group 4mm of the square consists of the cyclic group generated by the fourfold rotation 4+ containing the elements 1, 4+, 2, 4 and the reflections m10, m01, m11, [m_{1\bar1}] with mirror lines along the coordinate axes and the diagonals of the square (see Fig. 1.1.2.3[link]; the small black square in the centre represents the fourfold rotation point).

    [Figure 1.1.2.3]

    Figure 1.1.2.3 | top | pdf |

    Symmetry group 4mm of the square.

    The multiplication table of the group 4mm is

      1 2 4+ 4 m10 m01 m11 [m_{1\bar1}]
    1 1 2 4+ 4 m10 m01 m11 [m_{1\bar1}]
    2 2 1 4 4+ m01 m10 [m_{1\bar1}] m11
    4+ 4+ 4 2 1 m11 [m_{1\bar1}] m01 m10
    4 4 4+ 1 2 [m_{1\bar1}] m11 m10 m01
    m10 m10 m01 [m_{1\bar1}] m11 1 2 4 4+
    m01 m01 m10 m11 [m_{1\bar1}] 2 1 4+ 4
    m11 m11 [m_{1\bar1}] m10 m01 4+ 4 1 2
    [m_{1\bar1}] [m_{1\bar1}] m11 m01 m10 4 4+ 2 1

    This group is not abelian, because for example [{4}^+ \cdot {m}_{10} = {m}_{11}], but [{m}_{10} \cdot 4^+ = {m}_{1\bar1}].

The groups that are considered in crystallography do not consist of abstract elements but of symmetry operations with a geometric meaning. In the figures illustrating the groups and also in the symbols used for the group elements, this geometric nature is taken into account. For example, the fourfold rotation 4+ in the group 4mm is represented by the small black square placed at the rotation point and the reflection m10 by the line fixed by the reflection. To each crystallographic symmetry operation a geometric element is assigned which characterizes the type of the symmetry operation. The precise definition of the geometric elements for the different types of operations is given in Section 1.2.3[link] . For a rotation in three-dimensional space the geometric element is the line along the rotation axis and for a reflection it is the plane fixed by the reflection. Different symmetry operations may share the same geometric element, but these operations are then closely related, such as rotations around the same line. One therefore introduces the notion of a symmetry element, which is a geometric element together with its associated symmetry operations. In the figures for the crystallographic groups, the symbols like the little black square or the lines actually represent these symmetry elements (and not just a symmetry operation or a geometric element).

It is clear that for larger groups the multiplication table becomes unwieldy to set up and use. Fortunately, for many purposes a full list of all products in the group is actually not required. A very economic alternative of describing a group is to give only a small subset of the group elements from which all other elements can be obtained by forming products.

Definition. A subset [{\cal X} \subseteq {\cal G}] is called a set of generators for [{\cal G}] if every element of [{\cal G}] can be obtained as a finite product of elements from [{\cal X}] or their inverses. If [{\cal X}] is a set of generators for [{\cal G}], one writes [{\cal G} = \langle {\cal X} \rangle].

A group which has a finite generating set is said to be finitely generated.

Examples

  • (i) Every finite group is finitely generated, since [{\cal X}] is allowed to consist of all group elements.

  • (ii) A cyclic group is generated by a single element. In particular, the infinite cyclic group [(\bb Z, +)] is generated by [{\cal X} = \{ 1 \}], but also by [{\cal X} = \{ -1 \}].

  • (iii) The symmetry group 4mm of the square is generated by a fourfold rotation and any of the reflections, e.g. by [\{4^+, {m}_{10}\}], but also by two reflections with reflection lines which are not perpendicular, e.g. by [\{{m}_{10}, {m}_{11}\}].

  • (iv) The full symmetry group [m\bar3m] of the cube consists of 48 elements. It can be generated by a fourfold rotation [4^+_{100}] around the a axis, a threefold rotation [3^+_{111}] around a space diagonal and the inversion [\bar{1}]. It is also possible to generate the group by only two elements, e.g. by the fourfold rotation [4^+_{100}] and a reflection [m_{110}] in a plane with normal vector along one of the face diagonals of the cube.

  • (v) The additive group [({\bb Q}, +)] of the rational numbers is not finitely generated, because finite sums of finitely many generators [{{a_1}/{b_1}}, {{a_2}/{b_2}}, \ldots, {{a_n}/{b_n}}] have denominators dividing [b_1 \cdot b_2 \cdot \ldots \cdot b_n] and thus [{{1}/({1 + b_1 \cdot b_2 \cdot \ldots \cdot b_n})}] is not a finite sum of these generators.

Although one usually chooses generating sets with as few elements as possible, it is sometimes convenient to actually include some redundancy. For example, it may be useful to generate the symmetry group 4mm of the square by [\{2, m_{10}, m_{11}\}]. The element 2 is redundant, since [2 = (m_{10} m_{11})^2], but this generating set explicitly shows the different types of elements of order 2 in the group.








































to end of page
to top of page