1.3 Homomorphisms and isomorphisms

Group homomorphisms and isomorphisms are crucial tools in understanding relationships between groups. These mappings preserve group structure, with homomorphisms maintaining the operation and isomorphisms providing a complete structural equivalence.

Proofs for these mappings involve demonstrating operation preservation and, for isomorphisms, showing bijectivity. The First Isomorphism Theorem simplifies group structures, while kernels and images provide insights into the nature of homomorphisms and group relationships.

Group Homomorphisms and Isomorphisms

Group homomorphisms and isomorphisms

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• Group homomorphism maps elements preserving group operation $f: G \to H$ with $f(ab) = f(a)f(b)$ for all $a, b \in G$ (determinant function $GL_n(\mathbb{R}) \to \mathbb{R}^*$)
• Group isomorphism bijectively preserves group structure completely (complex exponential $(\mathbb{R}, +) \to (S^1, \cdot)$)
• Homomorphism examples include exponential map $(\mathbb{R}, +) \to (\mathbb{R}^+, \cdot)$
• Isomorphism examples include matrix representation of cyclic groups

Proofs of homomorphisms and isomorphisms

• Prove homomorphism:
  1. Demonstrate $f(ab) = f(a)f(b)$ for all $a, b \in G$
  2. Verify property for all element combinations
• Additional isomorphism proof steps:
  • Show injectivity (one-to-one)
  • Establish surjectivity (onto)
• Bijectivity proof techniques:
  • Construct inverse function
  • Compare group orders $|G| = |H|$ for finite groups
• Common strategies:
  • Direct proof approach
  • Contradiction method
  • Induction for infinite groups

First Isomorphism Theorem applications

• Theorem states $G/\ker(f) \cong \text{im}(f)$ for homomorphism $f: G \to H$
• Simplifies group structures and relates subgroups to normal subgroups
• Application process:
  1. Identify homomorphism
  2. Determine kernel and image
  3. Construct quotient group
  4. Establish isomorphism
• Examples: natural homomorphism group to quotient, $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$

Kernel and image in homomorphisms

• Kernel $\ker(f) = \{g \in G : f(g) = e_H\}$ forms normal subgroup of domain
• Image $\text{im}(f) = \{h \in H : h = f(g) \text{ for some } g \in G\}$ forms subgroup of codomain
• Injectivity equivalent to $\ker(f) = \{e_G\}$
• Surjectivity equivalent to $\text{im}(f) = H$
• Finite groups satisfy $|G| = |\ker(f)| \cdot |\text{im}(f)|$
• Kernel size indicates information loss, image size shows codomain coverage