Introduction to Group Theory

The theory of groups, an important part in present day mathematics, was established early in the nineteenth century in connection… Click here to read more

The concept of binary operations on a set is a generalization of the standard operations like addition and multiplication on… Click here to read more

1. Commutative Operation: A binary operation $$ * $$ over a set $$G$$ is said to be commutative if for… Click here to read more

Identity: A composition $$ * $$ in a set $$G$$ is said to admit of an identity if there exists… Click here to read more

A non-empty set $$G$$ together with at least one binary operation defined on it is called an algebraic structure. Thus… Click here to read more

An algebraic structure $$\left( {G, * } \right)$$ where $$G$$ is a non-empty set with a binary operation “$$ *… Click here to read more

A set $$G$$ with binary composition denoted multiplicatively is a group if (i) The composition is associative. (ii) For every… Click here to read more

If the commutative law holds in a group, then such a group is called an Abelian group or commutative group…. Click here to read more

Example 1: Show that the set of all integers …-4, -3, -2, -1, 0, 1, 2, 3, 4, … is… Click here to read more

Finite and Infinite Groups If a group contains a finite number of distinct elements, it is called a finite group…. Click here to read more

A binary operation in a finite set can completely be described by means of a table. This table is known… Click here to read more

Composition tables are useful in examining the following axioms in the manners explained below. Closure Property: If all the elements… Click here to read more

The identity element of a group is unique. The inverse of each element of a group is unique, i.e. in… Click here to read more

It is of common experience that a railway time table is fixed with the prevision of 24 hours in a… Click here to read more

Now here we are going to discuss a new type of addition, which is known as “addition modulo m” and… Click here to read more

Now here we are going to define another new type of multiplication, which is known as “multiplication modulo $$p$$.” It… Click here to read more