2.2 Kernels and Images of Homomorphisms

Group homomorphisms are key to understanding group structure. Kernels and images are crucial concepts that measure how well a homomorphism preserves group properties. They help us analyze the relationship between different groups and their subgroups.

Kernels tell us which elements map to the identity, while images show us what elements are "reached" by the homomorphism. These ideas are fundamental for the First Isomorphism Theorem, which connects kernels, images, and quotient groups in a powerful way.

Kernel and Image of Homomorphisms

Definitions and Properties

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• Kernel of group homomorphism φ: G → H consists of elements in G mapping to identity in H denoted as ker(φ) = {g ∈ G | φ(g) = eH}
• Image of group homomorphism φ: G → H comprises elements in H mapped by elements in G denoted as im(φ) = {h ∈ H | h = φ(g) for some g ∈ G}
• Kernel forms subgroup of domain G while image forms subgroup of codomain H
• Kernel measures failure of injectivity (ker(φ) = {eG} if and only if φ injective)
• Image measures failure of surjectivity (im(φ) = H if and only if φ surjective)
• Orbit-stabilizer theorem in group theory states |G| = |ker(φ)| · |im(φ)| for any group homomorphism φ

Examples and Applications

• Determinant function for matrix groups (kernel: special linear group SL(n,F))
• Sign function for permutation groups (kernel: alternating group An)
• Natural projection π: G → G/N (kernel: normal subgroup N)
• Exponential map exp: (ℝ,+) → (ℝ+, ·) (kernel: {0}, image: (0,∞))
• Modular arithmetic homomorphism φ: ℤ → ℤn (kernel: nℤ, image: ℤn)

Kernel as Normal Subgroup

Proof of Normality

• Demonstrate ker(φ) normal subgroup of G by proving closure under group operation, identity inclusion, inverse inclusion, and conjugation closure
• Closure under group operation: For a, b ∈ ker(φ), φ(ab) = φ(a)φ(b) = eHeH = eH, thus ab ∈ ker(φ)
• Identity element: φ(eG) = eH, therefore eG ∈ ker(φ)
• Inverse elements: For a ∈ ker(φ), φ(a⁻¹) = φ(a)⁻¹ = eH⁻¹ = eH, thus a⁻¹ ∈ ker(φ)
• Normality: For g ∈ G and k ∈ ker(φ), φ(gkg⁻¹) = φ(g)φ(k)φ(g⁻¹) = φ(g)eHφ(g)⁻¹ = eH, thus gkg⁻¹ ∈ ker(φ)
• Proof relies on homomorphism property φ(ab) = φ(a)φ(b) and conjugation in H preserving identity element

Importance and Applications

• Normality of kernel crucial for quotient group construction G/ker(φ)
• Enables First Isomorphism Theorem formulation
• Facilitates study of group structure through homomorphisms
• Allows classification of normal subgroups via kernels of homomorphisms
• Provides tool for analyzing group extensions and short exact sequences

Finding Kernel and Image

Calculation Techniques

• Evaluate φ(g) systematically for all g ∈ G to determine ker(φ) and im(φ)
• Utilize homomorphism property φ(ab) = φ(a)φ(b) to simplify calculations for large groups
• For cyclic groups, often sufficient to determine image of generator to characterize entire homomorphism
• In matrix groups, kernel often consists of scalar multiples of identity matrix mapping to identity
• For quotient groups, kernel of natural projection π: G → G/N precisely normal subgroup N

Common Homomorphisms and Their Kernels/Images

• Determinant function det: GL(n,F) → F* (kernel: SL(n,F), image: F*)
• Sign function sgn: Sn → {±1} (kernel: An, image: {±1})
• Abelianization map G → G/[G,G] (kernel: commutator subgroup [G,G], image: G/[G,G])
• Canonical projection Z → Z/nZ (kernel: nZ, image: Z/nZ)
• Inclusion map of subgroup H into G (kernel: {eH}, image: H)

Kernel, Image, and Isomorphism Theorem

First Isomorphism Theorem

• First Isomorphism Theorem states for group homomorphism φ: G → H, G/ker(φ) ≅ im(φ)
• Establishes fundamental connection between kernel, image, and quotient group structure
• Isomorphism given by map φ̄: G/ker(φ) → im(φ) defined by φ̄(gker(φ)) = φ(g)
• Implies every homomorphic image of group G isomorphic to quotient group of G
• Crucial for classifying all homomorphisms between groups and understanding subgroup structure

Applications and Generalizations

• Simplifies group calculations by reducing to quotient groups
• Proves other important results (Lagrange's Theorem, Third Isomorphism Theorem)
• Generalizes to other algebraic structures (rings, modules) with appropriate modifications
• Facilitates study of group extensions and short exact sequences
• Provides framework for analyzing normal subgroups and factor groups
• Enables construction of new groups from known ones via homomorphisms