## 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

From basic to higher mathematics

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

Suppose $$S$$ is a finite set having $$n$$ distinct elements. Then a one-one mapping of $$S$$ onto itself is called… Click here to read more

Two permutations $$f$$ and $$g$$ of degree $$n$$ are said to be equal if we have $$f\left( a \right) =… Click here to read more

If $$I$$ is a permutation of degree $$n$$ such that $$I$$ replaces each element by the element itself, $$I$$ is… Click here to read more

The products or composite of two permutations $$f$$ and $$g$$ of degree $$n$$ denoted by $$fg$$ is obtained by first… Click here to read more

If $$f$$ is a permutation of degree $$n$$, defined on a finite set $$S$$ consisting of $$n$$ distinct elements, by… Click here to read more

Let $$f$$ be a permutation on a set $$S$$. If a relation $$ \sim $$ is defined on $$S$$ such… Click here to read more

A permutation of the type \[\left( {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}& \cdots &{{a_{n – 1}}}&{{a_n}} \\ {{a_2}}&{{a_3}}&{{a_4}}& \cdots &{{a_n}}&{{a_1}} \end{array}} \right)\] is called… Click here to read more

Theorem 1: The product of disjoint cycles is commutative. Proof: Let $$f$$ and $$g$$ be any two disjoint cycles, i.e…. Click here to read more

The set $${P_n}$$ of all permutations on $$n$$ symbols is a finite group of order $$n{!}$$ with respect to the… Click here to read more

A permutation is said to be an even permutation if it can be expressed as a product of an even… Click here to read more

Suppose $$G$$ is a group and the composition has been denoted by multiplicatively, let $$a \in G$$. Then by closure… Click here to read more

If $$G$$ is a group and $$a$$ is an element of group $$G$$, the order (or period) of $$a$$ is… Click here to read more

Theorem 1: The order of every element of $$a$$ finite group is finite. Proof: Let $$G$$ be a finite group… Click here to read more

A group $$G$$ is called a cyclic group if, for some $$a \in G$$, every element $$x \in G$$ is… Click here to read more

Theorem 1: Every cyclic group is abelian. Proof: Let $$a$$ be a generator of a cyclic group $$G$$ and let… Click here to read more

Let $$G$$ be a group and $$H$$ any subset of $$G$$. Let $$a,b$$ be any two elements of $$H$$. Now… Click here to read more

The necessary and sufficient conditions for a subset of a group to be a subgroup are stated in the following… Click here to read more

Theorem 1: The intersection of two subgroups of a group $$G$$ is a subgroup of $$G$$. Proof: Let $${H_1}$$ and… Click here to read more

Theorem 1: Every subgroup of a cyclic group is cyclic. Proof: Let $$G = \left\{ a \right\}$$ be a cyclic… Click here to read more

If $$G$$ is a group, $$H$$ is a subgroup and $$a$$ is any element in $$G$$, then the set \[\left\{… Click here to read more

Let $$H$$ be a subgroup of group $$G$$. We know that no right coset of $$H$$ in $$G$$ is empty… Click here to read more

Theorem 1: If $$h \in H$$, then the right (or left) coset $$Hh$$ or $$hH$$ of $$H$$ is identical to… Click here to read more

Let $$H$$ be a subgroup of a group $$G$$. If the element $$a$$ of $$G$$ belongs to the right coset… Click here to read more

Lagrange’s Theorem The order of a subgroup of a finite group divisor of the order of the group. Proof: Let… Click here to read more

Let us consider the set of all complexes of a group $$G$$, which is nothing but a power set of… Click here to read more

By homomorphism we mean a mapping from one algebraic system with a like algebraic system which preserves structures. Definition Let… Click here to read more

Definition If $$f$$ is a homomorphism of a group $$G$$ into a $$G’$$, then the set $$K$$ of all those… Click here to read more

Here’s some examples of the concept of group homomorphism. Example 1: Let $$G = \left\{ {1, – 1,i, – i}… Click here to read more

Definition Let $$G$$ and $$G’$$ be any two groups with binary operation $$ \circ $$ and $$ \circ ‘$$, respectively…. Click here to read more

Theorem 1: If isomorphism exists between two groups, then the identities correspond, i.e. if $$f:G \to G’$$ is an isomorphism… Click here to read more

Theorem 1: Cyclic groups of the same order are isomorphic. Proof: Let $$G$$ and$$\,G’$$ be two cyclic groups of order… Click here to read more

Cayley’s Theorem: Every group is isomorphic to a permutation group. Proof: Let $$G$$ be a finite group of order $$n$$…. Click here to read more

Example 1: Show that the multiplicative group $$G$$ consisting of three cube roots of unity $$1,\omega ,{\omega ^2}$$ is isomorphic… Click here to read more

Let $$G$$ be an abelian group, the composition in $$G$$ being denoted multiplicatively. Let $$H$$ be any subgroup of $$G$$…. Click here to read more

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