Group theory

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

• 7.1 Chemistry 7.1.1 Sample Applications

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A group is called, if, here written for a lot together with a link of any two elements of this set as the following requirements are met:

• If you also still must swap the operands, so if it always holds true then there is an abelian group, also called a commutative group. ( Commutative )

Examples of abelian groups are

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

The very general definition of groups makes it possible to not only conceive sets of numbers with corresponding operations as groups, but also other mathematical objects with appropriate links that meet the above requirements. One such example is the set of rotations and reflections ( symmetry transformations ), is represented by the regular n-gon on itself, with the sequential execution of transformations as a shortcut ( dihedral group ).

Definition of a group

A group is a pair. Here is a set and a binary operation with respect. That is, by the image will be described. In addition, the following axioms for linking must be met in order to be designated as Group:

• Associativity: For all group elements, and the following applies:
• There is a neutral element, applies to the group for all elements.
• For each group element exists with an inverse element.

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

• Commutativity: and applies to all group members.

Otherwise, that is, when there are group elements, is for those who say the group of non- Abelian ( or non- commutative ).

Examples

Known examples of groups are:

• Klein's four group ( abelian )
• Symmetric group ( non- Abelian for n> 2)
• Alternating group ( non- Abelian for n> 3)
• Dihedral group ( non- Abelian for n> 2)
• Quaternion group ( non- Abelian )
• Trivial group: consists only of the identity element

A more detailed list, see the list of small groups.

Basic concepts of group theory

Order of a group

The cardinality ( cardinality ) of the carrier set of the group called the order of the group or group short order. For finite sets, this is simply the number of elements.

Order of elements

Produces a member of the group finally linked many times with itself, the neutral element 1, that is valid for a suitable n: it is called the order of the smallest such element. If no such exists, they say, that has infinite order. In both cases corresponds to the order of the element of the order of the subgroup generated by it.

On this basis, one can show that the order of each element of a finite group is finite and the group order divides ( Lagrange's theorem ).

The smallest number for which all group members is met simultaneously is called group exponent.

Subgroups

Is a subset of the underlying set of a group and is itself a group, it is called a subgroup of.

To this end, an important theorem (Theorem of Lagrange ): The order ( number of elements) of each subgroup of a finite group is a divisor of the order of the group, as applies. Specifically, is a prime number, then only the ( trivial ) subgroups (consisting of the identity element ) and itself

Cosets

If we define through to the group with a subgroup of the relation:

Obtained on an equivalence relation. The equivalence class to an element (that is, the class of all elements that are in the relation ), the amount

If on the other hand by a relation

Defined, then this is usually a different equivalence relation and the quantity of equivalent elements in now

Caused by right link of the elements of the element. It is referred to or called and right coset of to the item.

Induced link

Is a normal subgroup (see below ), then the left cosets are identical to the right cosets and one can define the cosets a link:

A non-empty subset of a group is exactly then a left or right coset of a subgroup, if always with elements is also the element.

Example

We consider the integers with addition as a group. Then the set of all integer multiples of 3, a sub-group. This results in three right cosets:

H H H 1 2 3 H = H 4 H = H 1 ... ......... -6 -5 -4 -3 -2 -1   0 1 2   3 4 5   6 7 8 ......... Since the amount of divisible by 3 numbers, the cosets are precisely the residue classes modulo 3 The table contains all integers such that no number appears twice, in a common column are respectively the numbers which when divided by three the same remainder.

Now one might be tempted here to expect only the cosets, ie modulo 3, and wonder if there is such a concept to each sub-group for any group. This leads to the following definition:

Normal subgroup

For each element is the left coset of equal to the right, that is, it is called a normal subgroup of.

In a commutative group every subgroup is a normal subgroup.

Factor group

The left cosets ( or even the right cosets ) with respect to a subgroup divide the group ( viewed as a set ) into disjoint subsets. If the subgroup even a normal subgroup, then every left coset is also a right coset and is now called only coset. For two given cosets the set of all possible products of an element of a coset with an element of the other coset is a coset again. One can therefore, be regarded as elements of a new group, the factor group the cosets.

In the construction of the factor group is ignored in particular that the cosets "really" are sets of group elements. The factor group is a kind of coarsened image of the original group.

The elements of the factor relative to the group of cosets for, and the link is given as follows

This definition is consistent, since the result of the choice of the elements of the cosets, g and h independently. This is called the link then well defined.

The defined with the cosets as elements and this linkage group is called the factor group of respect.

Cyclic groups

Does an element so that you can any other element as a power ( with an integer that may be negative ) can write, it is called a cyclic group and generating element.

Classification of finite simple groups

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

View

The properties of finite groups can be illustrated with the Rubik's cube, which has been widely used in academic classes since its invention, because the permutations of the corners and edges of the cube elements represent a visible and palpable example of a group.

Applications

Chemistry

The coordinates of the atoms of the molecules in their equilibrium conformation can be mapped onto itself by using symmetry operations ( mirroring, rotation, inversion, rotation-reflection ). The symmetry operations have the characteristics of groups of the dot groups, so-called. Also it can be shown that the group theory also applies to the symmetry of functions, including for wavefunctions in quantum mechanics.

Sample Applications

• Quantum Chemistry The computational complexity of quantum chemical calculations can be significantly reduced by using group theory, for example, has a Hamiltonian the same symmetry as its system.
• Furthermore, it is helpful to describe Salk ( symmetry -adapted linear combinations of atomic orbitals ), which has applications in molecular orbital theory and ligand field theory.
• Furthermore, the group theory is applied to the theory of conservation of orbital symmetry (see: Woodward -Hoffmann rules ).

• Spectroscopy Group theory is also used for infrared spectroscopy of importance, IR, Raman characteristics, presence of quadrupole and Octopolmoment can be read directly from the character table of a molecule.
• In NMR spectroscopy, protons, which can be mapped to each other by mirroring chemically equivalent and therefore result in the spectrum of the same chemical shift.

• Physical Properties A permanent electric dipole moment can only molecules of the point groups and
• Chirality / optical activity Molecules which do not have alternating axis, are chiral and therefore optically active, for example, bromo- chloro- iodo- methane
• Molecules which have a mirror line, are not optically active, even if they contain chiral centers, for example, meso compounds. Chiral catalysts in the enantioselective synthesis often contain ligands with symmetry, so that form defined complexes.

• Crystallography In crystallography group theory comes through the classification of crystal structures from the 230 possible space groups.

Physics

In quantum mechanics, symmetry groups are realized as groups of unitary or antiunitary operators. The eigenvectors of a maximal, abelian subgroup of these operators is distinguished from a physically important basis that belongs to states with well-defined energy or momentum or angular momentum or charge. For example, in form of solid state physics, the states in a crystal with an energy firm chosen a representation space of the symmetry group of the crystal.