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What is Group Theory? Properties (Axioms) and Applications

  • Neelam Tyagi
  • Oct 16, 2020
  • Updated on: May 31, 2021
What is Group Theory? Properties (Axioms) and Applications title banner

The importance of group theory was emphasized very recently when some physicists using group theory predicted the existence of a particle that had never been observed before, and described the properties it should have. Later experiments proved that this particle really exists and has those properties.—Irving Adler





The group theory is the most crucial ingredient in the present day of science, mathematics, statistics and computer science. It was ascertained in the nineteenth century in association with delivering solutions for algebraic expressions. 


In particular, the group was the set of all the permutations of the roots of an algebraic expression that exhibits the characteristics that the combination of any two of these permutations belongs to the set.


And later on, the belief was made generalized to the notion of an abstract group. However, an abstract group is the study of a set, with an operation defined on it. 


There are four primary sources in the development of group theory, that are (in the terms of author names and time of origin): 


  • Classical algebra (Lagrange, 1770) 

  • Number theory (Gauss, 1801)

  • Geometry (Klein, 1874), and  

  • Analysis (Lie, 1874; Poincaré and Klein, 1876)


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About Group Theory


The theory of group is essentially the study of groups where a group is a set equipped with specific binary operations. For example, the set of all the integers with addition.


If there are a finite number of components, the group is termed as a finite group, where the number of components is called the group order of the group. A subset of the group which is bounded under any of the group operations is known as Subgroups. 


More specifically, the formal definition of the group is;


  1. The group involves identity.

  2. The group includes inverse.

  3. The group holds an operation that is associative.

  4. The group is restricted under an operation.


In laymen terms, group theory is the study of the set of components present in a group, where a group is the acquisition of the components/ elements that are integrated together to perform some operations on them.


The group theory is the branch of abstract-algebra that is incurred for studying and manipulating abstract concepts involving symmetry.  It is the tool which is used to determine the symmetry. Also, symmetry operations and symmetry components are two fundamental and influential concepts in group theory.  


In core words, group theory is the study of symmetry, therefore while dealing with the object that exhibits symmetry or appears symmetric, group theory can be used for analysis.


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Why to look at Group Theory?


The simple answer is that Group Theory is the systematic and methodical study of symmetry, for an instance, when a physical quantity possess some sort of symmetry and so it undergoes any kind of symmetry operation in order to obtain a simplified description through accounting the outcomes of that symmetry.


Outcomes, obtained from group theory, could only be useful if and only if an individual understands them well enough to consider and deploy them up and provide users with a few basics insights as earlier as possible.


Group theory appears in the situation where the symmetry plays a significant part and its roots fall in Galois' study of the symmetries of the roots of polynomials.

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Properties of Group Under Group Theory


A group, G, is a finite or infinite set of components/factors, unitedly through a binary operation or group operation, that jointly meet the four primary properties of the group, i.e closure, associativity, the identity, and the inverse property. 


“Group theory is the natural language to describe the symmetries of a physical system.”


The operation (or formula) by virtue of which a group is determined is known as “Group Operation” and a set is treated as a group under that operation. Consider the elements A, B, C …….with binary operations amid A and B, and then notation “AB” form a group if 


  1. Closure: If A and B are two components in the group, G, then the product “AB” is also in “G”.

  2. Associativity: The defined multiplication is associative, therefore, for all A, B, C are in the group, “G”, then (AB) C= A (BC). 

    The image is highlighting the fundamental properties(axioms) of the group.Properties(Axioms) of the Group


  3. Identity: For any component, A, there also exists the identity element, I, such that IA= AI= A.


  4. Inverse: There should be an inverse of each component, so, for every component A under G, the set incorporates a component  B= A’ such that AA’= A’A= I. (from)  


Some other fundamental properties include;


  • A group is a monoid, where each of its components is invertible.

  • A group must comprise at least a single component, with the unique isomorphism, where the single-element group recognised as the trivial group.

  • The group is abelian if, for any A, B are elements of G, then AB=BA

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Applications of Group Theory


Group theory has multiple facets and various beneficial applications within and outside a broad width of science and mathematics, also groups mount in a number of supposedly impertinent entities. 


For example, they resemble in crystallography and quantum mechanics, in geometry and topology, in analysis and algebra, etc. Some of them are briefed below;


  1. In Physics, it plays an indispensable role in the determination of the selection rules for spectroscopic transitions in Atomic and Molecular Spectroscopy.

  2. In Chemistry, it is implemented to study and analyze the symmetries and the crystal structures of molecules, including many physical and chemical properties, and spectroscopic properties of the molecule. The group theory has turned out to be a standard and a powerful tool for studying molecular properties in the terms of molecular orbital theory.

  3. Group theory, being an eventual and powerful tool for symmetry, has an ultimate impact on research in robotics, computer vision, computer graphics and medical image analysis.

  4. In Mathematics, it can be used for classification of identical mathematical objects that possess symmetry, for example in geometric figures ( a circle is highly symmetric and invariant under any rotation) and in mathematical functions and operations.

  5. For the smooth data transmission, the concept of group, subgroups and cosets are used in Cryptography and Public Key algorithms.

  6. In recent researches, it has exploited in music screening as well.




Towards the end of the blog, we came to know that Group theory is the study of groups, where Groups are sets and furnished with a binary operation, such as multiplication, addition, or composition, that satisfies some fundamental basic properties. These properties are closure, associativity, identity, and inverse property.  


The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else or something else to the same thing.-James R. Newman


Groups serve as the building blocks of abstract algebra, they are more general and requisite that they appear in almost every branch of mathematics and the sciences.


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The concept of symmetry plays an important in group theory. Since group theory is the study of symmetry, therefore when an object or a system remains invariant under any transformation, then the object can be analyzed through the group theory.

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