Introduction to Group Theory

  • Introduction to Group Theory

    The theory of groups, an important part in present day mathematics, started early in nineteenth century in connection with the solutions of algebraic equations. Originally a group was the set of all permutations of the roots of an algebraic equation which has the property that combination of any two of these permutations again belongs to […]

  • Alternate Definition of a Group

    A set with binary composition denoted multiplicatively is a group if (i) The composition is associative. (ii) For every pair of elements , the equations and have unique solutions in . Proof: Binary operation implies that the set , under consideration is closed under the operation. Now to prove that the set is a group […]

  • Binary Operations

    The concept of binary operation on a set is a generalization of the standard operations like addition and multiplication on the set of numbers. For instance we know that the operation of addition (+) gives for ally two natural numbers another natural number , similarly the multiplication operation gives for the pair the number in […]

  • Types of Binary Operations

    1. Commutative Operation: A binary operation over a set is said to be commutative, if for every pair of elements , Thus addition and multiplication are commutative binary operations for natural numbers whereas subtraction and division are not commutative because, for and cannot be true for every pair of natural numbers and . For example […]

  • Identity and Inverse

    Identity: A composition in a set is said to admit of an identity if these exists an element such that Moreover, the element , if it exists is called an identity element and the algebraic structure is said to have an identity element with respect to. Examples: (1) If , the set of real numbers […]

  • Algebraic Structure

    A non-empty set together with at least one binary operation defined on it is called an algebraic structure. Thus if is a non-empty set and “” is a binary operation on , then is an algebraic structure. , , , are all algebraic structures. Since addition and multiplication are both binary operations on the set […]

  • Definition of Group

    An algebraic structure , where is a non-empty set with a binary operation “” defined on it is said to be a group; if the binary operation satisfies the following axioms (called group axioms). (G1) Closure Axiom. is closed under the operation , i.e., , for all . (G2) Associative Axiom. The binary operation is […]

  • Abelian Group or Commutative Group

    If the commutative law holds in a group, then such a group is called an Abelian group or Commutative group. Thus the group is said to be an Abelian group or commutative group if . A group which is not Abelian is called a non-Abelian group. The group is called the group under addition while […]

  • Examples of Group

    Example 1: Show that the set of all integers …,-4, -3, -2, -1, 0, 1, 2, 3, 4, ... is an infinite Abelian group with respect to the operation of addition of integers. Solution: Let us test all the group axioms for Abelian group. (G1) Closure Axiom. We know that the sum of any two […]

  • Order of a Group

    Finite and infinite Groups: If a group contains a finite number of distinct elements, it is called finite group otherwise an infinite group. In other words, a group is said to be finite or infinite according as the underlying set is finite or infinite. Order of a Group: The number of elements in a finite […]

  • Composition Table or Cayley Table

    A binary operation in a finite set can completely be described by means of a table. This table is known as composition table. The composition table helps us to verily most of the properties satisfied by the binary operations. This table can be formed as follows:   (i) Write the elements of the set (which […]

  • Group Tables

    The composition tables are useful in examining the following axioms in the manner explained below: Closure Property: If all the elements of the table belong to the set (say) then is closed under the Composition a (say). If any of the elements of the table does not belong to the set, the set is not […]