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. Inset 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. Geometric 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 group theory is, 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:
Closure: If x and y are two different elements in group G then x.y will also be a part of group G.
Associativity: If x, y, and z are the elements that are present in group G, then you get x. (y. z) = (x . y) . z.
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.
Identity: For any given element x in group G, there exists another element called I in group G in a way that x. I = I . x, wherein I refers to the identity element of 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, group theory studies the algebraic structures that are called groups. The concept of the group is a center to abstract algebra. The other well-known algebraic structures like the rings, fields and 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 group theory have influenced several parts of algebra. The linear algebraic groups and the Lie groups are the two branches of group theory that 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 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 ∈
G is unique. Also, prove that each of the elements x ∈
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 ∈ 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 = xex−1 = e
Similarly,
(x−1 * y-1) * (x * y) = e
Therefore, (xy)−1
= x−1
Y−1
Before learning group theory, one should have a clear knowledge of what a group is and how to define groups in sets. This will be helpful to understand the ideas of group theory in maths. A group is a collection of similar elements or objects that are combined together to perform specific operations. If any two objects are combined to produce a third element of the same set to meet four hypotheses namely closure, associativity, invertibility, and identity, they are called group axioms. Here is the definition, properties, and application of group theory.
The study of a set of elements present in a group is called a group theory in Maths. Its concept is the basic to abstract algebra. Algebraic structures like rings, fields, and vector spaces can be recognized as groups with axioms. The concepts and hypotheses of Groups are influenced throughout mathematics.
For Example, A group of numbers which are performed under multiplication operation.
Properties of Group Theory
If Dot(.) is an operation and G is a group, then the axioms of group theory is defined as;
Closure: If x, y are elements in a group, G, then x.y is also an element of G.
Associativity: If x, y and z are in group G, then x . (y . z) = (x . y) . z.
Invertibility: For every x in G, there is some y in G, such that; x. y = y . x.
Identity: For any element x in G, there is an element I in G, such that: x. I = I . x, where I is the identity of G.
The common example that satisfies these axioms is the addition of two integers, which is an integer. Hence, satisfies the closure property. The associative property is satisfied by the addition of integers. There is a zero identity in the group, which when added to any number, gives the original number. For every integer, there is an inverse, similarly, when added gives the result as zero. Hence, all the group axioms are satisfied in the addition operation of two integers.
Applications of Group Theory
The following are some of the important applications of Group Theory
If an object or a system property is invariant under the transformation, the object can be analyzed using group theory, because group theory is the study of symmetry.
Rubik’s cube can be solved using the algorithm of group theory.
Modeling of the crystals and the hydrogen atom are done using symmetry groups
Many fundamental laws of nature in Physics, Chemistry, and Material science use symmetry.
FAQs on Group Theory in Mathematics
1. What is group theory in mathematics?
Group theory is a branch of abstract algebra that studies algebraic structures called groups. A group is essentially a set of elements combined with a specific mathematical operation, which together must satisfy four fundamental properties. This framework is powerful because it allows mathematicians to analyse abstract concepts of symmetry and transformation that appear in various fields, from geometry to modern physics.
2. What are the four fundamental properties (axioms) of a group?
For a set 'G' combined with an operation '*' to be defined as a group, it must satisfy the following four axioms:
- Closure: For any two elements 'a' and 'b' in G, the result of their operation, a * b, must also be an element of G.
- Associativity: For any elements 'a', 'b', and 'c' in G, the equation (a * b) * c = a * (b * c) must hold true. The order of operations does not change the outcome.
- Identity Element: There must exist a unique element 'e' in G, known as the identity, such that for any element 'a' in G, a * e = e * a = a.
- Inverse Element: For each element 'a' in G, there must exist a corresponding element 'a⁻¹' in G, known as its inverse, such that a * a⁻¹ = a⁻¹ * a = e.
3. Can you provide a simple example of a group?
A classic and simple example of a group is the set of all integers (ℤ) under the operation of addition (+). It satisfies all four group axioms:
- Closure: Adding any two integers always results in another integer.
- Associativity: For any integers a, b, and c, (a + b) + c = a + (b + c).
- Identity Element: The identity element is 0, since a + 0 = a for any integer 'a'.
- Inverse Element: For any integer 'a', its inverse is '-a', because a + (-a) = 0.
4. Why is the set of integers under multiplication NOT a group?
The set of integers (ℤ) under the operation of multiplication fails to form a group because it violates the Inverse Element axiom. While it has closure, associativity, and an identity element (which is 1), not every element has a multiplicative inverse that is also an integer. For instance, the inverse of the integer 3 is 1/3, but 1/3 is not an integer. Because this property fails for most elements, the set is not a group.
5. What are some key applications of group theory?
Group theory has significant real-world applications because it is the mathematical language of symmetry. Key uses include:
- Physics: Describing the symmetries of physical laws and the properties of elementary particles in the Standard Model.
- Chemistry: Analysing molecular symmetry to understand crystal structures, spectroscopy, and chemical bonding.
- Cryptography: Forming the basis for secure public-key cryptography algorithms, such as the Diffie-Hellman key exchange.
- Computer Science: Used in algorithms for solving puzzles like the Rubik's Cube and in developing error-correcting codes.
6. How does group theory relate to the concept of symmetry?
Group theory is fundamentally the mathematical study of symmetry. The set of transformations that leave an object looking unchanged (like rotating a square by 90 degrees) can be structured as a group. The elements of the group are the symmetry operations themselves, and the group's operation is the act of applying one transformation after another. By studying the structure of this 'symmetry group', we can precisely understand all of an object's symmetrical properties.
7. What is the difference between an Abelian and a non-Abelian group?
The primary difference between these types of groups relates to the commutative property of their operation:
- An Abelian group is a group where the operation is commutative. This means for any two elements 'a' and 'b' in the group, a * b = b * a. The group of integers under addition is a common example.
- A non-Abelian group is one where the order of operation is important. There exist at least two elements 'a' and 'b' such that a * b ≠ b * a. The group of 3x3 invertible matrices under matrix multiplication is a classic non-Abelian example.
8. In a group, is it possible for an element to have more than one inverse?
No, it is not possible. A fundamental theorem in group theory states that every element in a group has a unique inverse. If an element 'a' were assumed to have two different inverses, it can be mathematically shown that those two inverses must be identical. This uniqueness is essential for the consistent and predictable structure that defines a group.
9. Who is considered the father of group theory?
The French mathematician Évariste Galois is widely regarded as the father of group theory. In the early 1830s, he used group-related concepts to solve a long-standing problem concerning the conditions under which a polynomial equation could be solved by radicals. His groundbreaking work laid the foundations for the entire field of abstract algebra.
