2.1 Definition and Examples of Homomorphisms

Group homomorphisms are functions that preserve the structure between groups. They map elements and operations from one group to another, maintaining key relationships like identity elements and inverses.

Homomorphisms play a crucial role in understanding group structures and relationships. They're used to classify groups, identify similarities, and study important concepts like normal subgroups, quotient groups, and group actions.

Group homomorphisms and their properties

Definition and key characteristics

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• Group homomorphism functions φ: G → H between two groups (G, *) and (H, •) preserve the group operation
• Defining property φ(a * b) = φ(a) • φ(b) holds for all elements a and b in G
• Preserve identity element mapping identity of G to identity of H
• Preserve inverses with φ(a⁻¹) = [φ(a)]⁻¹ for all a in G
• Kernel of homomorphism φ contains all elements in G mapping to identity element in H
• Image of homomorphism φ includes all elements in H mapped to by at least one element in G
• Homomorphisms categorized as injective (one-to-one), surjective (onto), or bijective (both)
  • Bijective homomorphisms called isomorphisms

Types of homomorphisms

• Trivial homomorphism maps every element of G to identity element of H
• Identity homomorphism from group to itself maps each element to itself
• Cyclic group homomorphism φ: ℤ → ℤn defined by φ(x) = x mod n
• Determinant function det: GL(n, ℝ) → ℝ* homomorphism from general linear group to multiplicative group of non-zero real numbers
• Exponential map exp: (ℝ, +) → (ℝ+, ×) defined by exp(x) = e^x homomorphism from additive group of real numbers to multiplicative group of positive real numbers
• Trace function tr: GL(n, ℝ) → (ℝ, +) homomorphism for matrix groups

Advanced properties

• Order of an element preserved or divided by homomorphism
  • Order of φ(a) divides order of a for all a in G
• Respect coset structure of groups
  • Map left (or right) cosets to left (or right) cosets
• First Isomorphism Theorem establishes relationship between group, homomorphic image, and kernel
• Used to define factor groups and quotient structures in group theory
• Lead to important concepts (normal subgroups, quotient groups, group actions)
• Crucial for classifying groups and identifying structural similarities

Examples of group homomorphisms

Common examples

• Trivial homomorphism: φ: ℤ → {0} mapping all integers to 0 in the trivial group
• Identity homomorphism: id: ℝ → ℝ mapping each real number to itself
• Modular arithmetic: φ: ℤ → ℤ6 defined by φ(x) = x mod 6
• Determinant: det: GL(3, ℝ) → ℝ* mapping 3x3 invertible matrices to their determinants
• Exponential function: exp: (ℝ, +) → (ℝ+, ×) defined by exp(x) = e^x
• Trace function: tr: M(2, ℝ) → ℝ mapping 2x2 matrices to the sum of their diagonal elements

Constructing homomorphisms

• Combine known homomorphisms on individual components for direct product groups
  • Example: φ: ℤ × ℤ → ℤ3 × ℤ4 defined by φ(a, b) = (a mod 3, b mod 4)
• Use polynomial functions for homomorphisms between additive groups
  • Example: φ: ℝ → ℝ defined by φ(x) = ax + b for fixed a, b ∈ ℝ
• Exploit structural similarities between groups to define homomorphisms
  • Example: φ: S3 → ℤ2 mapping even permutations to 0 and odd permutations to 1

Proving homomorphisms

Verification techniques

• Verify homomorphism property φ(a * b) = φ(a) • φ(b) for all elements a and b in domain group
• Check preservation of identity element by showing φ(e_G) = e_H
  • e_G and e_H represent identities of G and H respectively
• Confirm preservation of inverses by demonstrating φ(a⁻¹) = [φ(a)]⁻¹ for all a in G
• For finite groups, check homomorphism property on generating set of domain group
  • Example: Proving homomorphism for S3 by checking property on (12) and (123)
• Use counterexamples to disprove function as homomorphism if it fails required properties
  • Example: Showing f(x) = x² not a homomorphism from (ℝ, +) to (ℝ, ×) by f(1 + 2) ≠ f(1) × f(2)

Advanced proof strategies

• Utilize known properties of homomorphisms to support or refute claims
  • Preservation of subgroups: If H ≤ G, then φ(H) ≤ φ(G)
  • Preservation of normal subgroups: If N ◁ G, then φ(N) ◁ φ(G)
• Apply group theory theorems to simplify proofs
  • Example: Using Lagrange's theorem to prove order of φ(a) divides order of a
• Employ induction for homomorphisms involving infinite groups or recursively defined operations
  • Example: Proving φ(n) = n mod m homomorphism from (ℤ, +) to (ℤm, +) using induction on n

Homomorphisms and group structure preservation

Subgroup and normal subgroup preservation

• Homomorphisms map subgroups to subgroups
  • If H ≤ G, then φ(H) ≤ φ(G)
• Normal subgroups mapped to normal subgroups
  • If N ◁ G, then φ(N) ◁ φ(G)
• Kernel of homomorphism always normal subgroup of domain group
  • Ker(φ) ◁ G for any homomorphism φ: G → H
• Image of homomorphism subgroup of codomain group
  • Im(φ) ≤ H for any homomorphism φ: G → H

Order and structural relationships

• Order of element preserved or divided by homomorphism
  • Order of φ(a) divides order of a for all a in G
• Respect coset structure of groups
  • Left cosets mapped to left cosets: φ(aH) = φ(a)φ(H)
  • Right cosets mapped to right cosets: φ(Ha) = φ(H)φ(a)
• First Isomorphism Theorem establishes G/Ker(φ) ≅ Im(φ)
  • Provides powerful tool for studying group structures and relationships
• Homomorphisms used to define factor groups and quotient structures
  • Example: ℤ/nℤ as homomorphic image of ℤ under modulo n homomorphism

Applications in group theory

• Crucial for classifying groups and identifying structural similarities
  • Example: Using homomorphisms to show all cyclic groups of same order isomorphic
• Study of homomorphisms leads to important concepts
  • Normal subgroups as kernels of homomorphisms
  • Quotient groups as homomorphic images
  • Group actions as homomorphisms to permutation groups
• Homomorphisms provide framework for understanding symmetry and transformations
  • Example: Studying crystallographic groups through homomorphisms to orthogonal groups
• Used in representation theory to study abstract groups through linear transformations
  • Example: Character theory in finite group representations