## 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 is said to be commutative if for every pair of elements... Click here to read more

Identity: A composition in a set is said to admit of an identity if there exists an element such that... Click here to read more

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

An algebraic structure where is a non-empty set with a binary operation “” defined on it is said to be... Click here to read more

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

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

Two permutations and of degree are said to be equal if we have , . Example: If and are... Click here to read more

If is a permutation of degree such that replaces each element by the element itself, is called the identity permutation... Click here to read more

The products or composite of two permutations and of degree denoted by is obtained by first carrying out the operation... Click here to read more

If is a permutation of degree , defined on a finite set consisting of distinct elements, by definition is a... Click here to read more

Let be a permutation on a set . If a relation is defined on such that for some integrals ,... Click here to read more

A permutation of the type is called a cyclic permutation or a cycle. It is usually denoted by the symbol... Click here to read more

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

The set of all permutations on symbols is a finite group of order with respect to the composite of mappings... 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 is a group and the composition has been denoted by multiplicatively, let . Then by closure property etc., are... Click here to read more

If is a group and is an element of group , the order (or period) of is the least positive... Click here to read more

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

A group is called a cyclic group if, for some , every element is of the form , where is... Click here to read more

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

Let be a group and any subset of . Let be any two elements of . Now being a member... 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 is a subgroup of . Proof: Let and be any... Click here to read more

Theorem 1: Every subgroup of a cyclic group is cyclic. Proof: Let be a cyclic group generated by . Let... Click here to read more

If is a group, is a subgroup and is any element in , then the set is called the right... Click here to read more

Let be a subgroup of group . We know that no right coset of in is empty and any two... Click here to read more

Theorem 1: If , then the right (or left) coset or of is identical to , and conversely. Proof: Let... Click here to read more

Let be a subgroup of a group . If the element of belongs to the right coset , i.e. if... 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 , 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 is a homomorphism of a group into a , then the set of all those elements of which... Click here to read more

Here’s some examples of the concept of group homomorphism. Example 1: Let , which forms a group under multiplication and... Click here to read more

Definition Let and be any two groups with binary operation and , respectively. If there exists a one-one onto mapping... Click here to read more

Theorem 1: If isomorphism exists between two groups, then the identities correspond, i.e. if is an isomorphism and are respectively... Click here to read more

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

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

Example 1: Show that the multiplicative group consisting of three cube roots of unity is isomorphic to the group of... Click here to read more

Let be an abelian group, the composition in being denoted multiplicatively. Let be any subgroup of . If is an... Click here to read more

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