Group Theory
Before learning about group theory, let’s recall what is a group and how to define groups in sets. This will help you to get the introductory idea of group theory in maths. A group is a collection of elements or objects that are consolidated together to perform some operation on them. In set theory, we have been familiar with the topic of sets. If any two of its elements are combined through an operation to produce a third element belonging to the same set and meets the four hypotheses namely closure, associativity, invertibility and identity, they are called group axioms. In this article, you will the definition and meaning of group theory, axioms with proofs, properties and applications of group theory.
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Group Theory in Mathematics
Group theory is the study of a set of elements present in a group, in Maths. A group’s concept is fundamental to abstract algebra. Other familiar algebraic structures namely rings, fields, and vector spaces can be recognized as groups provided with additional operations and axioms. The concepts and hypotheses of Groups repeat throughout mathematics. Also, the rules of group theory have influenced several components of algebra. Also, we have an important theorem called the Lagrange theorem in group theory of mathematics.
For instance: A group of integers which are performed under multiplication operation. Geometric group theory in the branch of Mathematics is basically the study of groups that are finitely produced with the use of the research of the relationships between the algebraic properties of these groups and also topological and geometric properties of the spaces.
Group Theory Properties
Suppose Dot(.) is an operation and G is the group, then the axioms of group theory are defined as;
- Closure: If ‘x’ and ‘y’ are two elements in a group, G, then x.y will also come into 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 exists some ‘y’ in G, such that; x. y = y . x.
- Identity: For any element ‘x’ in G, there exists an element ‘I’ in G, such that: x. I = I . x, where ‘I’ is called the identity element of G.
The most common example, which satisfies these axioms, is the addition of two integers, which results in an integer itself. Hence, the closure property is satisfied. Also, the addition of integers satisfies the associative property. There exists an identity element name as zero in the group, which when added with any number, gives the original number. Also, for every integer, there exists an inverse, in such a way, when they are added gives zero as a result. So, all the group axioms are satisfied in the case of the addition operation of two integers.
Group Theory Axioms and Proof
Axiom 1: If G is a group that has a and b as its elements, such that a, b ∈ G, then (a × b)^{-1} = a^{-1} × b^{-1}
Proof:
To prove: (a × b) × b^{-1} × a^{-1}= I, where I is the identity element of G.
Consider the L.H.S of the above equation, we have,
L.H.S = (a × b) × b^{-1} × b^{-1}
=> a × (b × b^{-1}) × b^{-1}
=> a × I × a^{-1} (by associative axiom)
=> (a × I) × a^{-1} (by identity axiom)
= a × a^{-1} (by identity axiom)
= I (by identity axiom)
= R.H.S
Hence, proved.
Axiom 2: If in a group G, ‘x’, ‘y’ and ‘z’ are three elements such that x × y = z × y, then x = z.
Proof: Let us assume that x × y = z × y. (i)
Since ‘y’ is an element of group G, this implies there exist some ‘a’ in G with identity element I, such that;
y × a = I (ii)
On multiplying both sides of (i) by ‘a’ we get,
x × y × a = z × y × a
x × (y × a) = z × (y × a) (by associativity)
From eq.(ii);
a × I = c × I [using (ii)]
a = c (by identity axiom)
This is also known as cancellation law.
Group Theory Applications
The important applications of group theory are:
- Since group theory is the study of symmetry, whenever an object or a system property is invariant under the transformation, the object can be analyzed using group theory.
- The algorithm to solve Rubik’s cube works based on group theory.
- In Physics, the Lorentz group expresses the fundamental symmetry of many fundamental laws of nature.
Subgroup
Let (G, *) be a group structure and let S be a subset of G then S is said to be a subgroup of G if (S, *) is a group structure and if and only if it follows the properties given below.
(1) Binary Structure: ab ∈ S for every a, b ∈ S.
(2) Existence of Identity: Suppose e’ ∈ S such that e’a = a = ae’ for all a ∈ S.
(3) Existence of Inverse: For all a ∈ S, there exists a^{−1} ∈ S such that aa^{−1} = e = a^{−1}a.
Classes of Groups
The variety of groups being considered has steadily expanded from finite permutation groups and notable examples of matrix groups to abstract groups that may specify through a performance by generators and relations. Thus, we can define the following classes of groups.
- Permutation groups
- Matrix groups
- Transformation groups
- Abstract groups
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Number theory
The theory of algebraic equations
Geometry