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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.

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