Application of Group Theory in Discrete Mathematics
To learn the application of group theory in discrete mathematics, we will first learn about the group theory, which is described as follows:
In the present day of computer science, mathematics, science, and statistics, the group is described as the most crucial ingredient. The group theory is a type of natural language. In the 19th century, group theory was discovered to provide the solution of algebraic expressions. In modern algebra, the group theory can be described as the study of groups. According to laymen terms, the group theory can be described as the study of a set of components in a group. Here the group is a type of acquisition of the elements that are integrated together so that they can perform some operations on them.
The symmetries of a physical system can be easily described with the help of group theory. The group contains a binary operation and a set of elements. The group will be known as the finite group if it contains a finite number of components. The order of the group can be described by the number of components. There are also subgroups that can be described as a subset of a group that is bounded under any group.
We can apply the binary operation on two elements of the set that together satisfy certain axioms. That means if we combine two elements of the set with the help of binary operation, it will generate another element of the group and that meets the four hypotheses that can be identity, associative, closure and inverse. These four hypotheses can also be known as group axioms. With the help of a multiplication operation, a group of integers can be performed.
The group theory can also be referred to as the branch of abstract algebra. In discrete mathematics, the geometrical theory group can be referred to as the study of groups that are finitely produced by doing various researches of relationships between geometrical properties of space, topological and algebraic properties of groups. The group theory is a type of tool used to determine symmetry. In the group theory, there are two fundamental and influential concepts: symmetry components and symmetry operations.
Properties of Group Theory
In this, we are going to learn about the properties of group theory in discrete mathematics. Here the operation will be shown by the dot(.), and the group will be shown by G. The group G is an infinite or finite set of components, which is united with the help of binary operation or group operation. The operation which is used to determine a group is known as the Group Operation. Under that operation, a set will be treated as a group. In the group theory, the axioms are described as follows:
Suppose a group G contains the two elements 'a' and 'b'. The element will be closure if it contains the following relation:
(a, b) ∈ G ⇒ (a . b) ∈ G
Suppose a group G contains the elements 'a', 'b', and 'c'. The element will be associative if it contains the following relation:
(a . b) . c = a . (b . c) for all a, b, c in G
Suppose a group G contains the elements 'a', and another element 'e'. The element will be identified if it contains the following relation:
a . e = e . a for all a
Here, element 'e' can also be referred to as the identity element of G.
Suppose a group G contains two the elements 'a', 'b', and also contain an inverse element 'e'. The element 'b' will be referred to inverse of 'a' if it contains the following relation:
a . b = b . a = e
There is also one more way to represent the inverse of 'a', which is described as follows:
Applications of Group Theory
In discrete mathematics and science, group theory is used to study algebraic structures, which are known as groups. In abstract algebra, the group is the center. The groups are also seen by the other well known algebraic structures such as vector spaces, fields, rings. These other algebraic structures are endowed with axioms and additional operations. The several parts of algebra are influenced by the methods of group theory. The group theory has two branches: the lie group and the linear algebra group. These branches have experienced advances.
With the help of symmetry groups, various physical systems can be easily modeled, such as hydrogen and crystals atoms. So the group theory and the closely related theory are known as the representation theory because it contains a lot of important applications in the field of chemistry, physics, and material science.
The group theory has a lot of beneficial applications and multiple facets. Here groups also mount in various supposedly impertinent entities. For example: Group theory will resemble geometry and topology, crystallography, quantum mechanisms, analysis and algebra, and more places. Some of them are described as follows:
The group theory plays an indispensable role in Physics. In Atomic and molecular spectroscopy, the group theory is used to find out the selection rules for spectroscopic transitions.
With the help of implementing group theory in chemistry, we are able to analyze the crystal and symmetries structure of molecules. It is used to include the spectroscopic properties of a module and also include the various physical and chemical properties of modules. In the field of molecular orbital theory, the powerful and standard tool is group theory because it provides the ability to study molecular properties.
In the field of symmetry, the group theory can be referred to as a powerful tool. The graph theory provides an ultimate impact on research in medical image analysis, computer vision, robotics and computer graphics.
The group theory is able to provide a classification of identical mathematical objects, and they usually manage those objects that possess symmetry. The group theory will be useful in mathematical functions and operations and in geometric figures. For example: a circle is highly symmetric, which means if we rotate the circle, it will become invariant.
In public-key algorithms and cryptography, smooth data transmission is able to perform with the help of co-sets, groups, and subgroups.
According to recent researches, the group theory is also exploited in music screening.
Importance of Group Theory
In the group theory, we are going to study the algebraic objects, which are referred to as groups. We can use these groups in models, and it also studies the symmetries of a specific object. With the help of Galois Theory, we can explain the group theory. In this theorem, a new way will be found to attack the group to a polynomial. This polynomial will provide the proof of un-solvability of a fifth-degree polynomial.
The groups are mostly used in the topology, mainly in the algebraic topology, to capture certain invariants of space. Public-key cryptography also uses the group theory, which is used to efficiently carry out certain computations. The remainder of the integer will be modeled by the cyclic group, which is used to carrying out large computations.
Examples of Group Theory
The various examples of group theory are described as follows:
Suppose there is a group G. Now, we have to prove that the element e ∈ G is unique. We also have to prove that 'a' unique inverse is contained by each element a ∈ G. The syntax 'a-1' will be used to denote the unique inverse.
Here we will consider two identities elements e and e'. On the basis of the definition, we will get e' = e * e' = e.
Similarly, we will assume b and b' to be inverse of a. After that, we will get the following:
b = b * e
= b * (a * b')
= (b * a) * b'
= e * b'
Suppose group G contains two elements 'a' and 'b' and also the inverse of them as 'a-1' and 'b-1'. Now we have to find out the inverse of 'ab'.
Using the following way, we can describe the inverse of product of 'a' and 'b'.
a * b = a-1 * b-1
(a * b) * (a-1 * b-1)
= a (b * b-1) a-1
= aea-1 = e
(a-1 * b-1) * (a * b) = e
(ab)-1 = a-1 b-1