# Group theory

The group theory as a mathematical discipline examines the algebraic structure of groups .

A group clearly consists of the symmetries of an object or a configuration together with the link that is given by executing these symmetries one after the other. For example, the rotations of a regular corner in the plane with which the figure can be mapped onto itself form a group with elements. In order to grasp this concept in general, a concise and powerful definition has emerged: According to this, a group is a set together with a two-digit internal link (through which an element of this set is clearly assigned as a result to each ordered pair of elements), if this linkage is associative and there is a neutral element and an inverse for each element . For example, the set of whole numbers together with the addition forms a group. ${\ displaystyle n}$${\ displaystyle n}$

The systematic investigation of groups began in the 19th century and was triggered by specific problems, first of all by the question of the solvability of algebraic equations , and later by the investigation of geometric symmetries . Accordingly, the focus was initially on examining specific groups; It was not until the end of the 19th century that abstract questions were increasingly examined. Important contributions come from, among others, Évariste Galois and Niels Henrik Abel in algebra and Felix Klein and Sophus Lie in geometry. One of the outstanding mathematical achievements of the 20th century is the classification of all finite simple groups , i.e. the indivisible 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 in a uniform language it includes both geometric facts (movements of space, symmetries, etc.) and arithmetic rules (calculating with numbers, matrices, etc.) . Especially in algebra , the concept of a group is of fundamental importance: rings , solids , modules and vector spaces are groups with additional structures and properties. The methods and speech 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 basis and open up important applications.

## Access without mathematical requirements

Groups are used in mathematics to generalize computing with numbers. Correspondingly, a group consists of a set of things (e.g. numbers, symbols, objects, movements) and an arithmetic rule (a link, shown in this article as ), which indicates how to deal with these things. This calculation rule must comply with certain rules, the so-called group axioms, which are explained below. ${\ displaystyle *}$

One speaks of a group if the following requirements are met for a set together with a link between two elements of this set, here written as : ${\ displaystyle a * b}$

1. The combination of two elements of the set results in an element of the same set. (Isolation)
2. The brackets are irrelevant for the link, that is, it applies to all . ( Associative law )${\ displaystyle (a * b) * c = a * (b * c)}$${\ displaystyle a, b, c}$
3. There is one element in the set that has no effect on the connection, i.e. a - neutral element : for everyone .${\ displaystyle e}$${\ displaystyle *}$${\ displaystyle a * e = e * a = a}$${\ displaystyle a}$
4. For each element there is an inverse element with regard to the link, i.e. an - inverse element . This has the property when linking to give the neutral element: .${\ displaystyle a}$${\ displaystyle *}$ ${\ displaystyle a ^ {*}}$${\ displaystyle a}$${\ displaystyle a ^ {*} * a = a * a ^ {*} = e}$

Note: If the set speaks of several links, such as and , then there are several neutral and inverse elements, each matching the link. If it is clear from the context that only a certain link is meant, then one speaks briefly of the neutral element and the inverse element to without the link again explicitly mentioned. ${\ displaystyle *}$${\ displaystyle \ circ}$${\ displaystyle e}$${\ displaystyle a ^ {*}}$${\ displaystyle a}$

• If one can also swap the operands, that is, if it always applies, then there is an Abelian group , also called a commutative group. ( Commutative law )${\ displaystyle a * b = b * a}$

Examples of Abelian groups are

• the whole numbers with addition as a link and zero as a neutral element,${\ displaystyle \ mathbb {Z}}$${\ displaystyle +}$
• the rational numbers without zero with multiplication as a link and one as a neutral element. The zero must be excluded because it does not have an inverse element: "1/0" is not defined.${\ displaystyle \ mathbb {Q}}$${\ displaystyle \ cdot}$

The very general definition of groups makes it possible to understand not only sets of numbers with corresponding operations as groups, but also other mathematical objects with suitable links that meet the above requirements. One such example is the set of rotations and reflections (symmetry transformations) through which a regular n-gon is mapped onto itself, with the successive execution of the transformations as a link ( dihedral group ).

## Definition of a group

A group is a couple . There is a quantity and a two-digit link regarding . That is, this describes the figure . In addition, the following axioms must be fulfilled for the link so that it can be called a group: ${\ displaystyle (G, *)}$${\ displaystyle G}$${\ displaystyle *}$${\ displaystyle G}$${\ displaystyle * \ colon G \ times G \ to G, (a, b) \ mapsto a * b}$${\ displaystyle (G, *)}$

• Associativity : For all group elements,andapplies:${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle c}$${\ displaystyle (a * b) * c = a * (b * c).}$
• There is a neutral element , with all group items applies: .${\ displaystyle e \ in G}$${\ displaystyle a \ in G}$${\ displaystyle a * e = e * a = a}$
• For each group element one exists inverse element with .${\ displaystyle a \ in G}$ ${\ displaystyle a ^ {- 1} \ in G}$${\ displaystyle a * a ^ {- 1} = a ^ {- 1} * a = e}$

A group is called Abelian or commutative if the following axiom is also fulfilled: ${\ displaystyle (G, *)}$

• Commutativity : applies toall group elementsand.${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a * b = b * a}$

Otherwise, d. i.e., if there are group elements for which is, the group is called non-Abable . ${\ displaystyle a, b \ in G}$${\ displaystyle a * b \ neq b * a}$${\ displaystyle (G, *)}$

## Examples

Well-known examples of groups are:

A more detailed list can be found in the list of small groups .

## Basic concepts of group theory

### Order of a group

The cardinality (cardinality) of the set of carriers in the group is called the order of the group, or group order for short . For finite sets this is simply the number of elements. ${\ displaystyle | G |}$

### Subgroups

If a group is a subset of the carrier set and is itself a group, a subgroup of , is called a designation . ${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle (G, *)}$${\ displaystyle (H, *)}$${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle H \ leq G}$

An important theorem ( theorem of Lagrange ): The order of every subgroup of a finite group is a factor of the order of the group . If specifically a prime number , then only has the (trivial) subgroups (consisting of the neutral element) and itself. ${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle | G |}$${\ displaystyle G}$${\ displaystyle \ {e \}}$${\ displaystyle G}$

### Cyclic groups

If there is an element in such a way that each element can be written as a power (with an integer , which can also be negative), then one calls a cyclic group and generating element .${\ displaystyle G}$${\ displaystyle a}$${\ displaystyle a ^ {n}}$${\ displaystyle n}$${\ displaystyle G}$${\ displaystyle a}$

### Order of elements

If an element of the group, linked to itself finitely many times ( times), results in the neutral element 1, i.e. i.e. , if there is an with , the smallest such is called the order of the element . In this case one speaks of an element of finite order or torsion element . If there is no such thing , it is said to have infinite order . In both cases the order of the element corresponds to the order of the subgroup it creates . ${\ displaystyle a}$${\ displaystyle n}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle a ^ {n} = 1}$${\ displaystyle n> 0}$${\ displaystyle a}$${\ displaystyle n}$${\ displaystyle a}$

From Lagrange's theorem it follows: In a finite group , the order of each element is finite and a factor of the group order.

The smallest positive number that applies to each group element is called the group exponent . ${\ displaystyle n}$${\ displaystyle a ^ {n} = 1}$${\ displaystyle a}$

### Minor classes

Is defined on the group to a subgroup of the relation by ${\ displaystyle (G, *)}$${\ displaystyle (H, *)}$${\ displaystyle \ sim}$

${\ displaystyle a \ sim b \;: \ Longleftrightarrow \; \ exists \, h \ in H \ colon \, b = a * h}$,

obtained an equivalence relation on . The equivalence class for an element (i.e. the set of all elements that are related to) is the set ${\ displaystyle G}$${\ displaystyle a \ in G}$${\ displaystyle b}$${\ displaystyle a}$${\ displaystyle \ sim}$

${\ displaystyle \ {a * h \ mid h \ in H \}}$.

For this amount you write or . Since this set contains all elements of , which result from the fact that the element is linked with all elements from , it is called the left secondary class, alternative designation left remainder class, from after the element . ${\ displaystyle a * H}$${\ displaystyle aH}$${\ displaystyle G}$${\ displaystyle a}$${\ displaystyle H}$${\ displaystyle H}$${\ displaystyle a}$

On the other hand, if you have a relation through ${\ displaystyle a \ backsim b}$

${\ displaystyle a \ backsim b \;: \ Longleftrightarrow \; \ exists \, h \ in H \ colon \, b = h * a}$

defined, then this is generally a different equivalence relation and the set of elements to be equivalent in now ${\ displaystyle a}$${\ displaystyle G}$

${\ displaystyle \ {h * a \ mid h \ in H \}}$,

which is created by linking the elements to the right with the element . It is denoted by or and secondary legal class, alternative designation residual legal class, from after the element . ${\ displaystyle H}$${\ displaystyle a}$${\ displaystyle H * a}$${\ displaystyle Ha}$${\ displaystyle H}$${\ displaystyle a}$

Minor classes are used to prove Lagrange 's theorem, to explain the terms normal divisors and factor groups , and to study group operations.

### Double secondary classes

If two subgroups and are given, an equivalence relation is obtained through ${\ displaystyle K}$${\ displaystyle H}$

${\ displaystyle a \ sim b \;: \ Longleftrightarrow \; \ exists \, k \ in K, \, h \ in H \ colon \, b = k * a * h}$.

The equivalence class is ${\ displaystyle a \ in G}$

${\ displaystyle \ {k * a * h \ mid k \ in K, h \ in H \}}$

For this amount you write or calling it the - double coset to . ${\ displaystyle K * a * H}$${\ displaystyle KaH}$${\ displaystyle (K, H)}$${\ displaystyle a}$

### Normal divider

If for each element the left minor class is equal to the right one, i.e. H. , that's what a normal divisor of , is called a designation . ${\ displaystyle a \ in G}$${\ displaystyle H}$${\ displaystyle aH = Ha}$${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle H \ trianglelefteq G}$

In an Abelian group, every subgroup is a normal subgroup. The core of every group homomorphism is a normal divisor.

### Factor group

The left secondary classes (or also the right secondary classes) with respect to a subgroup divide the group (viewed as a set) into disjoint subsets. If the subgroup is even a normal subclass, then each left secondary class is also a right secondary class and from now on is only called a secondary class .

If is a normal divisor of , then a link can be defined on the set of the secondary classes: ${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle G / H}$

${\ displaystyle a_ {1} H * a_ {2} H: = \ left (a_ {1} * a_ {2} \ right) H}$

The link is well-defined , that is, it does not depend on the choice of representatives and in their secondary class. (If there is no normal divisor, then there are secondary classes with representatives that produce different results.) ${\ displaystyle a_ {1}}$${\ displaystyle a_ {2}}$${\ displaystyle H}$

Together with this induced link, the set of secondary classes forms a group, the factor group . The factor group is a kind of coarse image of the original group. ${\ displaystyle G / H}$

## Classification of the finite simple groups

A non-trivial group is called simple if it has no normal subgroups other than the trivial group and itself. For example, all groups of prime order are simple. The simple groups play an important role as the “building blocks” of groups. Since 1982 the finite simple groups have been fully classified. Each either belongs to one of the 18 families of finite simple groups or is one of the 26 exception groups , also known as sporadic groups .

## example

Rubik's Cube as an example of a finite non-Abelian group

Some properties of finite groups can be illustrated with the Rubik's Cube , which has been widely used in academic teaching since its invention, because the permutations of the corner and edge elements of the cube represent a visible and tangible example of a group.

## Applications

### chemistry

#### Point groups

The set of possible positions of the atoms of the molecules in their equilibrium conformation can be mapped onto itself with the help of symmetry operations (unit element, reflection, rotation, inversion, rotational reflection). The symmetry operations can be combined into groups, the so-called point groups .

#### Sample applications

• Physical properties
• Only molecules without any symmetry or symmetry of the point groups and and can have a permanent electric dipole moment .${\ displaystyle C_ {na}}$${\ displaystyle C_ {n}}$${\ displaystyle C_ {s}}$
• Chirality / optical activity
• Molecules that do not have a rotating mirror axis are chiral and therefore optically active, e.g. B. bromochloroiodomethane.${\ displaystyle S_ {n}}$
• Molecules that have a mirror axis are not optically active, even if they contain chiral centers, e.g. B. Meso compounds . Chiral catalysts in enantioselective synthesis often contain ligands with symmetry, so that defined complexes can be formed.${\ displaystyle C_ {2}}$

### physics

In quantum mechanics , symmetry groups are implemented as groups of unitary or anti-unit operators. The eigenvectors of a maximal Abelian subgroup of these operators characterize a physically important basis that belongs to states with well-defined energy or momentum or angular momentum or charge. For example, in solid-state physics, the states in a crystal with a fixed energy form a representation of the symmetry group of the crystal.

## history

The discovery of group theory is attributed to Évariste Galois , who attributed the solvability of algebraic equations by radicals (in today's terminology) to the solvability of their Galois group . Galois' work was not published posthumously until 1846. The concept of a group already played an implicit role in Lagrange ( Réflexions sur la résolution algébrique, 1771) and Gauss ( Disquisitiones Arithmeticae, 1801).

In the last quarter of the 19th century, group theory became a central component of mathematics , primarily through Felix Klein's Erlanger program and the theory of continuous transformation groups developed by Sophus Lie , as well as Poincare's and Klein's work on automorphic functions . Poincaré's well-known quote, “Les mathématiques ne sont qu'une histoire des groupes.” (Mathematics is just a history of groups) dates from 1881.

An abstract definition of groups can be found for the first time in 1854 by Arthur Cayley :

"A set of symbols all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group . These symbols are not in general convertible [commutative] but associative, it follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor [left or right], the effect is simply to reproduce the group. " ${\ displaystyle 1, \ alpha, \ beta, \ ldots,}$

It was not until 1878 that the first works on abstract group theory appeared. Cayley proved that every finite group is isomorphic to a group of permutations and noted in the same paper that it is easier to think of groups as abstract groups rather than groups of permutations. In 1882 Dyck first defined groups using generators and relations.

## literature

• Pavel S. Alexandroff: Introduction to group theory. German, Frankfurt 2007, ISBN 978-3-8171-1801-4 .
• Hans Kurzweil, Bernd Stellmacher: Theory of finite groups. An introduction. Springer, Berlin 1998, ISBN 3-540-60331-X .
• Thorsten Camps among others: Introduction to combinatorial and geometric group theory. Heldermann, Lemgo 2008, ISBN 978-3-88538-119-8 .
• Oleg Bogopolski: Introduction to group theory. European Math. Soc., Zurich 2008, ISBN 978-3-03719-041-8 .
• Stephan Rosebrock: Illustrative Group Theory - A Computer-Oriented Geometrical Introduction. Springer Spectrum, 2020, ISBN 978-3-662-60786-2 .