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Group theory as a mathematical discipline studies the algebraic structure of groups.

Graphically, a group consists of the symmetries of an object or configuration together with that linkage which is given by the succession of these symmetries. For example, the rotations of a regular -corner in the plane, by which the figure can be mapped onto itself, form a group with elements. A concise and powerful definition has emerged to capture this concept in general terms: According to it, a group is a set together with a two-digit inner linkage (by which each ordered pair of elements is uniquely assigned an element of that set as a result) if this linkage is associative and there is a neutral element as well as an inverse to each element. For example, the set of integers together with addition also forms a group.

The systematic study of groups began in the 19th century and was triggered by concrete problems, first by the question of the solvability of algebraic equations, later by the study of geometric symmetries. Accordingly, the study of concrete groups was initially in the foreground; it was not until the end of the 19th century that abstract problems were increasingly studied. Important contributions were made by Évariste Galois and Niels Henrik Abel in algebra and Felix Klein and Sophus Lie in geometry, among others. One of the outstanding mathematical achievements of the 20th century is the classification of all finite simple groups, that is, the indecomposable building blocks of all finite groups.

The great importance of group theory for many areas of mathematics and its applications results from its generality, because it includes in a unified language both geometrical facts (movements of space, symmetries, etc.) and arithmetical rules (arithmetic with numbers, matrices, etc.). Especially in algebra the notion of group is of fundamental importance: rings, solids, moduli and vector spaces are groups with additional structures and properties. Methods and ways of speaking of group theory therefore pervade many areas of mathematics. In physics and chemistry, groups appear wherever symmetries play a role (e.g. invariance of physical laws, symmetry of molecules and crystals). For the investigation of such phenomena, group theory and the closely related representation theory provide the theoretical foundations and open up important applications.

Access without mathematical prerequisites

Groups are used in mathematics to generalize arithmetic with numbers. Accordingly, a group consists of a set of things (e.g., numbers, symbols, objects, motions) and a computational rule (a conjunction, represented in this article as ∗ ) that specifies how to deal with these things. This computational rule must satisfy certain rules, called group axioms, which are explained below.

A set is said to be a group if, for a set together with a conjunction of each two elements of that set, written here as the following requirements are fulfilled:

1. The conjunction of two elements of the set in turn gives an element of the same set. (Completeness)
2. For the linkage, the bracketing is irrelevant, that is, it holds for all . (Associative Law)
3. There is an element in the set that does nothing with respect to the link, that is, a ∗ -neutral element: for all .
4. For every element there is, with respect to the link, an inverse element, that is, a ∗ {\displaystyle -inverse element . This has the property of yielding the neutral element when concatenated with : .

Note that if there are multiple links on the set, such as ∗ and , then there are multiple neutral and inverse elements, each matching the linkage. If it is clear from the context that only one particular linkage is meant, then one briefly speaks of the neutral element and the inverse element to without explicitly mentioning the linkage again.

• If, in addition, the operands may be interchanged, i.e., if always holds, then we have an abelian group, also called a commutative group. (Commutative Law)

Examples of abelian groups are

• the integers with addition as the link and zero as the neutral element,
• the rational numbers without zero with multiplication as the link and one as the neutral element. The zero must be excluded here, since it has no inverse element: "1/0" is not defined.

The very general definition of groups makes it possible to take not only sets of numbers with corresponding operations as groups, but also other mathematical objects with suitable linkages that satisfy the above requirements. Such an example is the set of rotations and reflections (symmetry transformations) by which a regular n-corner is mapped onto itself, with the successive execution of the transformations as a linkage (Dieder group).

Definition of a group

A group is a pair . Here a set and ∗ a two-digit link with respect to . That is, this describes the mapping ∗ Moreover, the following axioms for the linkage must be satisfied for called a group:

• Associativity: For all group elements , and holds that
• There exists a neutral element with which for all group elements holds: .
• For every group element there exists an inverse element with .

A group is called abelian or commutative if in addition the following axiom is satisfied:

• Commutativity: for all group elements and , .

Otherwise, i.e., if there exist group elements for which , the group is called nonabelian.