then usually the cyclic group , the dihedral group , or the group of all affine mappings on are symmetry groups on which can be motivated by music theory (c.f. Mazzola, 1990, 2002).

That kind of constructions over and many other objects in discrete mathematics can be best described in the notion of discrete structures. The mathematical tool for working with symmetry operations is the notion of group actions, which will be introduced later.

In general discrete structures are objects constructed as

• subsets, unions, products of finite sets,
• mappings between finite sets,
• bijections, linear orders on finite sets,
• equivalence classes on finite sets,
• vector spaces over finite fields, etc.

For example it is possible to describe graphs, necklaces, designs, codes, matroids, switching functions, molecules in chemistry, spin-configurations in physics, or objects of local music theory as discrete structures.

As was indicated above, often the elements of a discrete structure are not simple objects, but they are themselves classes of objects which are considered to be equivalent. Then each class collects all those elements which are not essentially different. For instance, in order to describe mathematical objects we often need labels, but for the classification of these objects the labelling is not important. Thus all elements which can be derived by relabelling of one labelled object are collected to one class.

For example, a labelled graph is usually described by its set of vertices and its set of edges . (An edge connects exactly two different vertices of the graph.) If the graph has vertices, then usually and is a subset of the set of all -subsets of . Then belongs to if the two vertices with labels and are connected by an edge of the graph. An unlabelled graph is the set of all graphs which can be constructed by relabelling a labelled graph.

Besides relabelling, also naturally motivated symmetry operations give rise to collect different objects to one class of essentially not different objects. This is for instance the case when we describe different partitions of or different canons on .

The process of classification of discrete structures provides more detailed information about the objects in a discrete structure. We distinguish different steps in this process:

Step 1:

Determine the number of different objects in a discrete structure.

Step 2:

Determine the number of objects with certain properties in a discrete structure.