 # Group Theory in Mathematics

## What is Group Theory?

Group theory in mathematics refers to the study of a set of different elements present in a group. A group is said to be a collection of several elements or objects which are consolidated together for performing some operation on them. In the set theory, you have been familiar with the topic of sets. If any two of the elements of a set are combined through an operation for producing a third element that belongs to the same set and that meets the four hypotheses that are the closure, the associativity, the invertibility, and the identity, they are referred to as group axioms. A group of integers is performed under the multiplication operation. Geometrical group theory according to the branch of mathematics refers to the study of the groups which are finitely produced by using the research of the relationships between the different algebraic properties of these groups and the topological and the geometric properties of space. In this article, we will learn about what is group theory, what are the applications of group theory in mathematics, and look at some group theory examples.

### Properties of Group Theory

Let us learn about group theory math properties.

Consider dot (.) to be an operation and G to be a group. The axioms of the group theory are defined in the following manner:

1. Closure:

If x and y are two different elements in the group G then x.y will also be a part of the group G.

1. Associativity:

If x, y, and z are the elements that are present in group G, then you get x. (y. z) = (x . y) . z.

1. Invertibility:

For every element x in the group G, there exists some y in the group G in a way that; x. y = y . x.

1. Identity:

For any given element x in the group G, there exists another element called I in the group G in a way that x. I = I . x, wherein I refers to the identity element of the group G.

### Applications of Group Theory

Let us now look at what are the applications of group theory in mathematics.

In Mathematics and abstract algebra, the group theory studies the algebraic structures that are called groups. The concept of the group is a center to the abstract algebra. The other well-known algebraic structures like the rings, fields, and the vector spaces are all seen as the groups that are endowed with the additional operations and axioms. Groups recur throughout when it comes to mathematics, and the methods of the group theory have influenced several parts of algebra. The linear algebraic groups and the Lie groups are the two branches of the group theory which have experienced advances and are the subject areas in their own ways.

Several physical systems like the crystals and the hydrogen atom can be modeled by the symmetry groups. Hence the group theory and the closely related theory called the representation theory to have several important applications in the fields of physics, material science, and chemistry. The group theory is also the center of the public key cryptography.

### Group Theory Examples

Let us look at some of the group theory examples.

Example 1

Let G be a group. Prove that the element e $\in$ G is unique. Also, prove that each of the element x $\in$ G consists of a unique inverse which is denoted by $x^{-1}$.

Solution

Consider e and e’ to be the identities.

According to the definition, you get e' = e * e' = e.

Similarly, consider y and y' to be the inverses of x.

Then, you would get

y = y * e

= y * (x * y’)

= (y * x) * y’

= e * y’

= y’

Example 2

Consider x, y $\in$ G having the inverses $x^{-1}$. and $y^{-1}$. respectively. Determine the inverse of xy.

Solution

The inverse of the product of x and y is given as follows:

x * y = $x^{-1}$ * $y^{-1}$

You have (x * y) * ($x^{-1}$ * $y^{-1}$)

= x (y * $y^{-1}$) $x^{-1}$

= xe$x^{-1}$ = e

Similarly,

($x^{-1}$ * y-1) * (x * y) = e

Therefore, $(xy)^{-1}$ = $x^{-1}$  $y^{-1}$

1. What is the importance of group theory?

Group theory studies the algebraic objects known as groups that can be used for model and hence study about the symmetries of a specific object. Group theory is widely used in pure math as well. An example of this is the Galois Theory. Evariste Galois had found a way for attaching the group to a polynomial that led to his proof of the unsolvability of fifth-degree polynomial, and an entirely new area of math. In topology, especially the algebraic topology, the groups are used for the capture of certain invariants of space.

Group theory is also useful in the public-key cryptography for efficiently carrying out certain computations. The cyclic groups are used for modeling the remainders of the integers that is useful for carrying out larger computations.

2. What are the applications of group theory?

1. In Physics, the Lorentz group expresses the fundamental symmetry of several known fundamental laws of nature. The special theory of the relativity, which is Maxwell's field equations in the electromagnetism and the Dirac equation in the theory of electron are all invariant under the Lorentz transformations.

2. Modern particle physics is related to the symmetry principles and through the application of the group theory, the existence of different particles was predicted before they were observed experimentally.

3. In chemistry, the symmetry of a particular molecule provides you with the information on the energy levels of the orbitals, the orbitals symmetries, the transitions that occur between different energy levels, and even the bond order that is calculated using the group theory.