International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 24
Section 1.1.2. Basic properties of groups^{a}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands 
Although groups occur in innumerable contexts, their basic properties are very simple and are captured by the following definition.
Definition. Let be a set of elements on which a binary operation is defined which assigns to each pair of elements the composition . Then , together with the binary operation , is called a group if the following hold:
In most cases, the composition of group elements is regarded as a product and is written as or even instead of . An exception is groups where the composition is addition, e.g. a group of translations. In such a case, the composition is more conveniently written as .
Examples
If a group 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 . A group with infinitely many elements is called an infinite group.
For a group element , its order is the smallest integer such that is the identity element. If there is no such integer, then is said to be of infinite order.
The group operation is not required to be commutative, i.e. in general one will have . However, a group in which for all is said to be a commutative or abelian group.
The inverse of the product of two group elements is the product of the inverses of the two elements in reversed order, i.e. .
A particularly simple type of groups is cyclic groups in which all elements are powers of a single element . A finite cyclic group of order n can be written as . For example, the rotations that are symmetry operations of an equilateral triangle constitute a cyclic group of order 3.
The group 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 .
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 and of the column labelled by is the product . 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
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 m_{10} 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 . For a rotation in threedimensional 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 is called a set of generators for if every element of can be obtained as a finite product of elements from or their inverses. If is a set of generators for , one writes .
A group which has a finite generating set is said to be finitely generated.
Examples

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 . The element 2 is redundant, since , but this generating set explicitly shows the different types of elements of order 2 in the group.