Skip to main content

This Week's Best Picks from Amazon

Please see more curated items that we picked from Amazon here .

Introduction to Groups

  • Understand the definition of a group.
  • Learn the properties and axioms of groups.
  • Explore examples of groups.

Definition of a Group

A group is a set \(G\) equipped with a binary operation \(*\) that combines any two elements \(a, b \in G\) to form another element \(a * b \in G\), and satisfies the following axioms:

  1. Closure: For all \(a, b \in G\), \(a * b \in G\).
  2. Associativity: For all \(a, b, c \in G\), \((a * b) * c = a * (b * c)\).
  3. Identity Element: There exists an element \(e \in G\) such that for all \(a \in G\), \(a * e = e * a = a\).
  4. Inverse Element: For each \(a \in G\), there exists an element \(b \in G\) such that \(a * b = b * a = e\), where \(e\) is the identity element.

Group Properties and Axioms

  • Closure: The binary operation \(*\) must be closed in the group \(G\).
  • Associativity: The group operation must be associative.
  • Identity Element: There must be an identity element \(e\) in the group such that \(a * e = e * a = a\) for all \(a \in G\).
  • Inverse Element: Every element \(a\) in the group must have an inverse \(b\) such that \(a * b = b * a = e\).

Examples of Groups

  • Integers under Addition: The set of integers \(\mathbb{Z}\) with the operation of addition is a group. The identity element is \(0\), and the inverse of \(a\) is \(-a\).
  • Non-zero Real Numbers under Multiplication: The set of non-zero real numbers \(\mathbb{R}^*\) with the operation of multiplication is a group. The identity element is \(1\), and the inverse of \(a\) is \(\frac{1}{a}\).
  • Symmetric Group: The set of all permutations of a finite set \(S\) forms a group under the operation of composition. This group is called the symmetric group and is denoted by \(S_n\) for a set of \(n\) elements.

Exercises

  • Verify Group Properties:
    • Show that the set of integers \(\mathbb{Z}\) with addition is a group by verifying the closure, associativity, identity, and inverse properties.
    • Show that the set of non-zero real numbers \(\mathbb{R}^*\) with multiplication is a group by verifying the closure, associativity, identity, and inverse properties.
  • Examples of Groups:
    • Find the identity element and inverse for the set of integers modulo \(n\), denoted by \(\mathbb{Z}_n\), with addition modulo \(n\).
    • Show that the set of \(2 \times 2\) invertible matrices with real entries forms a group under matrix multiplication.

Summary

In this lesson, we introduced the concept of a group, defined its properties and axioms, and explored several examples of groups. Understanding the definition and properties of groups is fundamental to studying more advanced topics in abstract algebra.

Popular posts from this blog

Eigenvalues and Eigenvectors

Lesson Objectives ▼ Understand the definition of eigenvalues and eigenvectors. Learn how to compute eigenvalues and eigenvectors. Explore the characteristic equation. Interpret eigenvalues and eigenvectors in transformations. Understand diagonalization and its applications. Lesson Outline ▼ Definition of Eigenvalues and Eigenvectors Characteristic Equation Diagonalization Examples Definition of Eigenvalues and Eigenvectors In a linear transformation, an eigenvector is a nonzero vector that only changes by a scalar factor when the transformation is applied. For a square matrix \( A \), an eigenvector \( \mathbf{v} \) and its corresponding eigenvalue \( \lambda \) satisfy: \[ A\mathbf{v} = \lambda \mathbf{v} \] \( A \) is an \( n \times n \) matrix. \( \mathbf{v} \neq 0 \) is an eigenvector. \( \lambda \) is a scalar eigenvalue. Characteristic Equation ...
Read more

Sets and Binary Operations

Lesson Objectives ▼ Understand the concept of a set and its basic operations. Learn about binary operations and their properties. Explore examples of binary operations. Lesson Outline ▼ Introduction to Sets Basic Set Operations Binary Operations Introduction to Sets A set is a collection of distinct objects, considered as an object in its own right. For example, the set of natural numbers \( \{1, 2, 3, \ldots\} \). Sets can be represented using curly braces, e.g., \( \{a, b, c\} \). Basic Set Operations Union: The union of sets \(A\) and \(B\) is the set of elements that are in \(A\), \(B\), or both. Denoted as \(A \cup B\). Intersection: The intersection of sets \(A\) and \(B\) is the set of elements that are in both \(A\) and \(B\). Denoted as \(A \cap B\). Difference: The difference of sets \(A\) and \(B\) is the set of elements that are in \(A\) but not in \(B\). Denoted as \(A...
Read more

Inner Product Spaces

Lesson Objectives ▼ Understand the definition of inner product spaces. Compute norms, angles, and distances in inner product spaces. Explore the concept of orthogonality. Learn the Gram-Schmidt orthogonalization process. Use inner products for projections and least squares approximation. Lesson Outline ▼ Definition of Inner Product Spaces Norms, Angles, and Orthogonality Gram-Schmidt Orthogonalization Projections and Least Squares Examples Definition of Inner Product Spaces An inner product space is a vector space with an operation called the **inner product**, which measures similarity between vectors. For a real vector space, the inner product of two vectors \( \mathbf{u}, \mathbf{v} \) is: \[ \langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 + u_2 v_2 + \dots + u_n v_n \] For a complex vector space, the inner product is: \[ \langle \mathbf{u}, \mathbf{v} \ran...
Read more