Definition of a Group in Abstract Algebra

Definition of a Group

• A group is a set of elements, called group elements, together with a binary operation (usually represented by multiplication or addition) that satisfies four properties:
  1. Closure: The result of combining any two group elements is always an element in the group.
  2. Associativity: The order in which group elements are combined does not affect the result.
  3. Identity: There exists an identity element (usually denoted as e) that does not change the result when combined with any group element.
  4. Inverse: For each group element, there exists an inverse element that, when combined, results in the identity element.

Properties of Groups

• Commutativity: If the group operation is commutative, the group is called an Abelian group.
• Finite and Infinite Groups: A group can have a finite or infinite number of elements.
• Order of a Group: The number of elements in a group, denoted by |G|.
• Order of an Element: The smallest positive integer n such that an element raised to the power of n equals the identity element.

Types of Groups

• Finite Groups: Groups with a finite number of elements, e.g., symmetric groups, dihedral groups.
• Infinite Groups: Groups with an infinite number of elements, e.g., additive and multiplicative groups of integers.
• Abelian Groups: Commutative groups, e.g., integers under addition, integers modulo n under multiplication.
• Non-Abelian Groups: Non-commutative groups, e.g., symmetric groups, general linear groups.

Group Operations

• Multiplicative Notation: Group operation denoted by multiplication, e.g., G = {a, b, c}, a ∘ b = c.
• Additive Notation: Group operation denoted by addition, e.g., G = {a, b, c}, a + b = c.
• Modular Arithmetic: Group operation performed modulo a certain number, e.g., addition and multiplication modulo n.

Important Theorems

• Lagrange's Theorem: The order of any subgroup divides the order of the group.
• Sylow's Theorems: Theorems describing the existence and properties of Sylow p-subgroups.

Definition of a Group

• A group consists of a set of elements, known as group elements, and a binary operation (usually represented by multiplication or addition) that satisfies four properties.

Properties of a Group

• A group must satisfy four properties: closure, associativity, identity, and inverse.
• Closure: The result of combining any two group elements is always an element in the group.
• Associativity: The order in which group elements are combined does not affect the result.
• Identity: There exists an identity element (usually denoted as e) that does not change the result when combined with any group element.
• Inverse: For each group element, there exists an inverse element that, when combined, results in the identity element.

Properties of Groups

• Commutativity: A group is called an Abelian group if the group operation is commutative.
• A group can have a finite or infinite number of elements.
• Order of a Group: The number of elements in a group, denoted by |G|.
• Order of an Element: The smallest positive integer n such that an element raised to the power of n equals the identity element.

Types of Groups

• Finite Groups: Groups with a finite number of elements, e.g., symmetric groups, dihedral groups.
• Infinite Groups: Groups with an infinite number of elements, e.g., additive and multiplicative groups of integers.
• Abelian Groups: Commutative groups, e.g., integers under addition, integers modulo n under multiplication.
• Non-Abelian Groups: Non-commutative groups, e.g., symmetric groups, general linear groups.

Group Operations

• Multiplicative Notation: Group operation denoted by multiplication, e.g., G = {a, b, c}, a ∘ b = c.
• Additive Notation: Group operation denoted by addition, e.g., G = {a, b, c}, a + b = c.
• Modular Arithmetic: Group operation performed modulo a certain number, e.g., addition and multiplication modulo n.

Important Theorems

• Lagrange's Theorem: The order of any subgroup divides the order of the group.
• Sylow's Theorems: Theorems describing the existence and properties of Sylow p-subgroups.