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Group Homomorphisms and Isomorphisms

  • Understand the definition and properties of group homomorphisms.
  • Learn about isomorphisms and their characteristics.
  • Explore examples of group homomorphisms and isomorphisms.

Definition of Group Homomorphisms

A group homomorphism is a function between two groups that preserves the group operation. If \( G \) and \( H \) are groups with operations \( * \) and \( \cdot \) respectively, a function \( \phi: G \to H \) is a homomorphism if for all \( a, b \in G \),

\( \phi(a * b) = \phi(a) \cdot \phi(b) \).

Properties of Group Homomorphisms

  • Identity Preservation: A homomorphism maps the identity element of the first group to the identity element of the second group. If \( e \) is the identity in \( G \), then \( \phi(e) \) is the identity in \( H \).
  • Inverse Preservation: A homomorphism maps the inverse of an element in the first group to the inverse of its image in the second group. If \( a \in G \), then \( \phi(a^{-1}) = \phi(a)^{-1} \).

Definition of Isomorphisms

An isomorphism is a bijective homomorphism. If \( \phi: G \to H \) is an isomorphism, then \( \phi \) is both a homomorphism and a bijection. Two groups \( G \) and \( H \) are said to be isomorphic if there exists an isomorphism between them.

Notation: If \( G \) and \( H \) are isomorphic, it is denoted as \( G \cong H \).

Properties of Isomorphisms

  • Structure Preservation: Isomorphisms preserve the group structure, meaning that the group operation, identity element, and inverse elements are preserved.
  • Bijective: Isomorphisms are both injective (one-to-one) and surjective (onto).

Examples of Group Homomorphisms and Isomorphisms

  • Example 1: Consider the groups \( (\mathbb{Z}, +) \) and \( (\mathbb{Z}_n, +) \). The function \( \phi: \mathbb{Z} \to \mathbb{Z}_n \) defined by \( \phi(a) = a \mod n \) is a homomorphism.
  • Example 2: The function \( \phi: (\mathbb{R}, +) \to (\mathbb{R}^+, \cdot) \) defined by \( \phi(x) = e^x \) is an isomorphism. This shows that the group of real numbers under addition is isomorphic to the group of positive real numbers under multiplication.

Exercises

  • Identify Homomorphisms: Given the groups \( (\mathbb{Z}, +) \) and \( (\mathbb{Z}_6, +) \), determine if the function \( \phi: \mathbb{Z} \to \mathbb{Z}_6 \) defined by \( \phi(a) = a \mod 6 \) is a homomorphism.
  • Isomorphism Verification: Show that the groups \( (\mathbb{Q}, +) \) and \( (\mathbb{Q}^+, \cdot) \) are isomorphic by finding an appropriate isomorphism.
  • Homomorphism Properties: Verify that the function \( \phi: (\mathbb{Z}, +) \to (\mathbb{Z}, +) \) defined by \( \phi(a) = 2a \) is a homomorphism, and determine if it is also an isomorphism.

Summary

In this lesson, we explored the concepts of group homomorphisms and isomorphisms, including their definitions, properties, and examples. Understanding these fundamental ideas is crucial as they provide insights into the relationships between different groups and their structures.

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