Review of Group theory and its application

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Introduction 1.Fields
Being little more than a precisely specified collection of mathematical objects.A set is not particularly helpful in and of itself on the other hand a set become extremely helpful when one or more alternative (like addition and multiplication) define for its constituents.The operations will have a specially "rich" algebraic structure if they completely with well known arithmetic laws (such as distributivity, commutativity, and associativity) fields are those sets that have the richest algebraic structure.Real numbers (rationals and irrational numbers) complex numbers (numbers of the type a+ib where a and b are real numbers and i 2 =-1) and rational numbers (fractions a/b where a and b are positive or negative whole integers) are common examples of fields, each of these its own symbol Q for the rational, R for the real and C for the complex numbers the term fields in its algebraic sense is quite different from its use in other context.Such as vector fields in mathematics are magnetic fields in physics other languages avoid this conflict in terminology for example a field in the algebraic sense is called a corps in frech and a korper in German both words meaning "body".

Structural Axioms
The table displays the fundamental principles or axioms for addition and multiplication.A set that complies with each of the ten principles is referred to as a field.A commutative ring is a set that satisfies the commutative law of multiplication (Axioms 8).when axioms 1 -9 hold and there are no proper divisors of zero.(that is whenever ab=0 either a=0 or b=0) a set is called an integral domain.For example the set of integers {…-2,-1, 0, 1, 2…} is a commutative ring with unity but it is not a field.Because axioms 10 fails when only axiom 8 fails a set is known as a division or skew field.

Prime Factorization
The 19th century work on number theory also laid the ground work for some other essential ideas of modern algebra, especially in relation to efforts to extend the prime factorization theorem outside the natural numbers.According to this theorem each natural number could be expressed uniquely as the product of its prime components one should not be surprised the to find the name of Gauss in this context and his classical investigations.Gauss was led to the factorization properties of the numbers of the type a+ib (a and b are integers and i= square root of √-1) sometime called Gaussian integers.In doing so Gauss not only demonstrated how to empty complex numbers to solve an ordinary integer problem.A noteworthy accomplishment in and of itself but he also paved the way for a through examinations of unique subdomains of complex numbers.In 1832 Gauss proved that Gaussian integers satisfied a generalised version of prime factorization theorem.Where the prime factors had to especially defined in this domain.In the 1840 the German mathematician Ernst Eduard Kummer extended these results to other even more general domains of complex numbers such as numbers of the form a+b where  2 =n for n a fixed integers are numbers of the form a+b where  n =1, ⧧1 and n>2.Although Kummer did proof interesting results it finally turned out that the prime factorization theorem was not valid in such general domains.The following example illustrates the problem.

Research Methodology
In contemporary algebra, group theory refers to the study process that involves systems made up of a set of elements and inary operations that can be applied to two of the elements in the sets that together meet specific axioms.These stipulates that the group must have an identity elements be closed under the operations fulfill the associative law and have an inverse for each elements A group is referred to as commutative or abelian, if it also satisfies the commutative law the set of integers under addition where the identity element is 0 and inverse is the negative of a positive number are vice versa, is an abelian group.Groups are vital to modern algebra their basic structure can be found in many mathematical phenomina.Group can be found in geometry, representing phenomina such as symmetry and certain types of transformations.Group theory has applications in chemistry physics and computer science and even puzzles like rubik's cube can be represented using group theory.

Methods and Material
We have attempted to identify the greatest outcome and recent research on the fundamentals of objectives in this paper.Both the current and initial work on these guidelines have been completed.I am currently using group theory to address algebraic mathematical difficulties during the renaissance mathematicians discovered roots extractions and coefficients for general polynomials of degree 3 and 4 which provided analogues for the quadratic formula.

Structures in Moden Algebra
Fields, rings and groups.We will investigate the definitions and some example for the time we do not proof anything without it that will come in next chapters.When we look in the depth of this structure.
We will denote different type of notations for various kinds of numbers.The set of natural numbers {1, 2, 3,…} is denoted by N. The set of integers {-2,-1, 0, 1, 2,…} is denoted by Z.The set of rational numbers are represented by Q the set of all positive , negative number including 0 is denoted by R and the complex numbers which is written in the form x+iy is denoted by C in which first term are real numbers and second terms are imaginary numbers.
Thus such type of structures is very useful in modern algebra.

The Role of Group Theory in Moden Cryptography
The aim of this study is to explore the Role of Group Theory in Modern Cryptography.Group theory is a branch of mathematics that has played a significant role in modern cryptography, particularly in the development of public-key cryptosystems, digital signatures, and elliptic curve cryptography.Groups are sets of elements together with a binary operation that satisfies four axioms: closure, associativity, identity, and inverse.Group theory is used to study the properties of objects that remain unchanged when transformed by certain operations, and it has been applied to a wide range of fields, including physics, chemistry, and cryptography.Public-key cryptosystems allow two parties who have never met before to communicate securely over an insecure channel.They rely on the fact that some mathematical problems are difficult to solve, such as factoring large numbers into their prime factors.Public-key cryptosystems use two keys, a public key and a private key.The public key can be freely distributed, while the private key is kept secret.Messages are encrypted using the recipient's public key and can only be decrypted using their private key.The security of public-key cryptosystems is based on the difficulty of certain mathematical problems, and the underlying mathematical structure is often a group that satisfies the axioms of group theory.Digital signatures are used to ensure the authenticity and integrity of digital documents.They rely on a mathematical function that takes as input the document and the signer's private key and produces a signature that can be verified using the signer's public key.Elliptic curve cryptography is a form of public-key cryptography that uses elliptic curves instead of the integers modulo a large prime as its underlying mathematical structure.It can be concluded that, group theory has played a crucial role in the development of modern cryptography.Its use in public-key cryptosystems, digital signatures, and elliptic curve cryptography has enabled secure communication over insecure channels, secure digital transactions, and secure data storage.As the field of cryptography continues to evolve, group theory will undoubtedly remain an essential tool for ensuring the security of encrypted data.Group theory is a powerful branch of mathematics that provides a framework for understanding and analyzing the behavior of sets of objects under certain operations.It has many important applications in mathematics, science, and engineering, and is an essential tool for solving problems in a wide range of fields.One of the strengths of group theory is its ability to provide a unified language and set of tools for studying a wide range of phenomena.For example, the symmetries of a geometric object can be described using group theory, and the behavior of algorithms can be analyzed using the theory of permutation groups.Group theory also provides a powerful tool for classifying and categorizing objects based on their properties.Another important aspect of group theory is its connection to other areas of mathematics.Many concepts and techniques from algebra, geometry, and topology have been developed using group theory, and many important results in these fields rely on the theory of groups.For example, the study of Lie groups and their representations is an important area of mathematics that has connections to physics, geometry, and topology.Despite its many applications and successes, there are still many open questions and challenges in group theory.One of the most important is the so-called "inverse Galois problem", which asks whether every finite group can be realized as the Galois group of some finite extension of the rational numbers.Another important problem is the classification of infinite simple groups, which is still an open question.In recent years, there have been many exciting developments in group theory, including new techniques for studying the structure of groups, new connections to other areas of mathematics and science, and new applications in emerging fields such as quantum computing and machine learning.Group theory continues to be a vibrant and active area of research, with many opportunities for new discoveries and breakthroughs in the future.Cryptography is the study of techniques for secure communication in the presence of adversaries.It is an ancient field that has been used for centuries to protect messages and sensitive information.Today, cryptography plays a critical role in modern society, with applications in areas such as internet security, banking, and digital privacy.The goal of cryptography is to design algorithms and protocols that ensure the confidentiality, integrity, and authenticity of information transmitted between parties.Confidentiality means that the information is protected from unauthorized access, integrity means that the information has not been tampered with or altered, and authenticity means that the information comes from a trusted source.One of the most important concepts in cryptography is encryption, which is the process of transforming plaintext, or unencrypted information, into cipher text, or encrypted information.The process of encryption typically involves a key, which is a piece of information used to control the encryption and decryption process.The key can be a password, a cryptographic key, or a combination of both.There is two main types of encryption: symmetric-key encryption and public-key encryption.In symmetric-key encryption, the same key is used for both encryption and decryption.This key is typically kept secret and is only known to the sender and the receiver of the encrypted message.Examples of symmetric-key encryption algorithms include the Data Encryption Standard (DES) and the Advanced Encryption Standard (AES).In public-key encryption, two different keys are used for encryption and decryption.One key, called the public key, is known to everyone and is used for encryption.The other key, called the private key, is keptand is only known to the owner of the public key.The private key is used for decryption.Public-key encryption was first introduced by Whitfield Diffie and Martin Hellman in 1976 and has since become a widely used cryptographic technique.Examples of public-key encryption algorithms include the RSA algorithm and the Elliptic Curve Cryptography (ECC) algorithm.Another important concept in cryptography is digital signatures, which are used to ensure the authenticity of digital documents or messages.A digital signature is created by applying a cryptographic hash function to the document or message and then encrypting the result using the signer's private key.The digital signature can then be decrypted using the signer's public key to verify the authenticity of the document or message.Cryptography also involves the study of protocols for secure communication between parties.One of the most important protocols is the Transport Layer Security (TLS) protocol, which is used to secure communication between web browsers and web servers.TLS is based on public-key encryption and provides a secure channel for transmitting sensitive information such as passwords and credit card numbers.Cryptography has many important applications in modern society.One of the most important is internet security, where cryptography is used to protect sensitive information transmitted over the internet, such as passwords, credit card numbers, and personal information.Cryptography is also used in digital rights management (DRM) systems to protect copyrighted material from unauthorized copying or distribution.In addition, cryptography is used in the banking industry to protect financial transactions and to ensure the integrity of data transmitted between banks and financial institutions.Cryptography is also used in government and military applications to protect classified information and to ensure the security of communication between government agencies and military units.Despite its many successes, cryptography is a constantly evolving field, with new challenges and threats emerging all the time.One of the most important challenges is the threat of quantum computing, which could potentially break many of the cryptographic algorithms currently in use.To address this threat, researchers are developing new cryptographic algorithms

Cyclic Groups
This section will introduce cyclic groups: definitions, characteristics that are inherent and come along with the definition, and examples that concretely illustrate the concept.Definition (cyclic groups) cyclic groups are a specific type of group that is generated by one element.The element  that generates the group is called the group generator, and the group is often denoted as.By definition, is the set of invertible elements , where each x can be acquired through repeatedly performing the binary operation of the group to  or its inverse.

Cyclic Groups in Number Theory
This section will look at the application of cyclic groups to number theory.The reliance of many number theory lemmas upon the characteristics of prime numbers allows for the close relationship between cyclic groups (in particular, simple cyclic groups) and number theory.Cyclic group manifests in another area of mathematics -number theory.To begin with, number theory ties closely within cyclic simple groups, as the cyclic group ℤ (the set of integers modulo n) comes from the very concept of modulus within number theory.The close relation is even furthered when used in applications like the Chinese Remainder Theorem in number theory, from which it is possible to determine the group structure of the group ℤ for any ∈ℤ by expanding it out into a prime factorized form.

Sylow Group
When studying group theory one notice almost immediately that groups of prime power orders are great significance, with Cauchy's , Lagrange's and Sylow 's theorems being three good examples of this.The study of these so called pgroups, where p is prime number, can for example be used to give a clear understanding of other groups as being compositions of different p-groups.The sylow theorems are collection of theorems named after the Norwegian mathematician Pater Ludwig sylow (1872) that give detailed information about the number of subgroups of fixed order that given finite groups contains, the sylow theorems forms a fundamental part of finite group theory and have important application of finite simple group.Further this theorem also asserts of Lagrange's theorem.

Vector Space in Group Theory
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars.Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field.The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms.Real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars: real coordinate space or space.Vector spaces generalize Euclidean vectors, which allow modelling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction.The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces.This provides a concise and synthetic way for manipulating and studying equations.Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space.This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic).A vector space is finite-dimensional if its dimension is a natural number.Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal.Finite-dimensional vector spaces occur naturally in geometry and related areas.Infinite-dimensional vector spaces occur in many areas of mathematics.For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension.Many vector spaces that are considered in mathematics are also endowed with other structures.This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras.This is also the case of topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces.

Basic Knowledge of Ring
In algebra, ring theory is the study of rings-algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and identities.Commutative are much better understood than noncommutative ones.Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics.Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to.For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra.Similarly, Fermat's Last Theorem is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.
Noncommutative rings are quite different in flavour, since more unusual behavior can arise.While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (nonexistent) 'noncommutative spaces'.This trend started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups.It has led to a better understanding of noncommutative rings, especially noncommutative rings.For the definitions of a ring and basic concepts and their properties, see Ring (mathematics).The definitions of terms used throughout ring theory may be found in Glossary of ring theory.

Commutative Rings
A ring is called commutative if its multiplication is commutative.Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to formalize properties of the integers.Commutative rings are also important in algebraic geometry.In commutative ring theory, numbers are often replaced by ideals, and the definition of the prime ideal tries to capture the essence of prime numbers.Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility.Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers.Euclidean domains are integral domains in which the Euclidean algorithm can be carried out.Important examples of commutative rings can be constructed as rings of polynomials and their factor rings.Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring.

Algebraic Geometry
Algebraic geometry is in many ways the mirror image of commutative algebra.This correspondence started with Hilbert's Nullstellensatz that establishes a one-to-one correspondence between the points of an algebraic variety, and the maximal ideals of its coordinate ring.This correspondence has been enlarged and systematized for translating (and proving) most geometrical properties of algebraic varieties into algebraic properties of associated commutative rings.Alexander Grothendieck completed this by introducing schemes, a generalization of algebraic varieties, which may be built from any commutative ring.More precisely, the spectrum of a commutative ring is the space of its prime ideals equipped with Zariski topology, and augmented with a sheaf of rings.These objects are the "affine schemes" (generalization of affine varieties), and a general scheme is then obtained by "gluing together" (by purely algebraic methods) several such affine schemes, in analogy to the way of constructing a manifold by gluing together the charts of an atlas.

Non Commutative Rings
Noncommutative rings resemble rings of matrices in many respects.Following the model of algebraic geometry, attempts have been made recently at defining noncommutative geometry based on noncommutative rings.Noncommutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules.A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces.Examples of noncommutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings.

Applications
 Cryptography: In cryptography group theory is used in different ways public key, cryptosystem, digital signature and elliptic curve cryptography.In public key cryptography allows to for parties in which two keys public and private key are used public key is known by everyone but private key is kept secret .Message are encrypted using the recipients public key but decrypted by only private key.The security of public key cryptosystem is used on mathematical problems.The most widely used public key are RSA.In which group satisfies the four axioms of group theory.
 Physics: The group theory has a important in physics in atomic and molecular spectroscopy.The group theory is used to the selection rules for spectroscopic transitions. Chemistry: In chemistry the group theory is used to analysis the crystal and symmetries structures of molecules.It is used to spectroscopic properties of module and also the physical and chemical properties of module. in the field of orbital theory group theory is very powerful tool because it provide the ability to study molecular properties  Computer: In the field of symmetry group theory is used as a powerful tool.The group theory play a important role on research in medical image analysis computer vision, robotics and computer graphics. Symmetry Analysis: Group theory provides a powerful tool for analyzing symmetry in objects and systems.In chemistry, for instance, it helps in understanding molecular structure and properties by analyzing the symmetry of molecules. Particle Physics: In particle physics, group theory is used to describe the fundamental forces of nature.For example, the Standard Model of particle physics relies heavily on group theory, particularly groups, to describe the symmetries of particles and their interactions. Crystallography: Group theory is essential in crystallography for understanding the symmetries of crystals.
The study of crystal symmetries helps in predicting and interpreting various properties of crystals, such as their optical and mechanical behavior. Error-Correcting Codes: Group theory plays a role in the design and analysis of error-correcting codes, which are used to detect and correct errors in digital communication and storage systems.Groups, particularly permutation groups, are used to define the symmetries and structures of these codes. Topology: Group theory has applications in topology, particularly in the study of fundamental groups and homology groups.These algebraic structures help in classifying and distinguishing different types of topological spaces. Number Theory: Group theory is used in number theory, particularly in the study of modular arithmetic and algebraic structures like rings and fields.Group theory concepts are also essential in the study of elliptic curves, which have applications in cryptography and integer factorization.

Conclusion
The short introduction to group theory is exposing group structure .Group can be mapped one onto another and how groups can operate on ensembles .it is very useful in algebra of material .In physics matrices are related to mappings between vector space or Hilbert spaces and often endomorphism i.e. mapping of a vector space onto itself .Group theory is used to solved cryptography problems .In which message are encrypted and decrypted by group properties.

Disclosure of conflict of interest
No conflict of interest to be disclosed.