Key Concepts of Homomorphisms

Why This Matters

Homomorphisms are the fundamental tools for understanding how algebraic structures relate to one another—they're the "structure-preserving maps" that let you compare groups, identify hidden symmetries, and break complex structures into simpler pieces. When you're tested on this material, you're really being asked to demonstrate that you understand why certain mappings preserve group operations, how kernels and images reveal structural information, and what it means for two groups to be "essentially the same."

The concepts here—kernels, images, isomorphisms, quotient groups—form the backbone of modern algebra and connect directly to geometric symmetries you'll encounter throughout the course. Don't just memorize definitions; know what each concept tells you about the relationship between groups and why the First Isomorphism Theorem is considered one of the most powerful results in group theory.

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The Foundation: What Homomorphisms Are

A homomorphism is fundamentally a map that respects algebraic structure—it translates the "grammar" of one group into another without losing meaning. The key insight is that the group operation in the domain gets carried faithfully to the codomain.

Definition of a Homomorphism

• Structure-preserving map—a homomorphism $f: G \to H$ satisfies $f(a \cdot b) = f(a) \cdot f(b)$ for all elements $a, b$ in the domain group
• Applies across algebraic structures—while we focus on groups, homomorphisms exist between rings, vector spaces, and other structures with the same preservation property
• Foundation for all comparison—every concept in this guide builds on this single defining property

Homomorphism Properties

• Identity preservation—homomorphisms automatically satisfy $f(e_G) = e_H$, a consequence of the structure-preserving property, not an additional requirement
• Inverse preservation—for any element $g$, we have $f(g^{-1}) = f(g)^{-1}$, maintaining the inverse structure
• Composition closure—the composition of two homomorphisms is itself a homomorphism, allowing you to chain structure-preserving maps

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Measuring Homomorphisms: Kernels and Images

The kernel and image are your diagnostic tools—they tell you exactly how much information a homomorphism "loses" and what portion of the codomain it "hits." Together, they completely characterize the behavior of any homomorphism.

Kernel of a Homomorphism

• Definition—the kernel $\ker(f) = \{g \in G \mid f(g) = e_H\}$ is the set of all elements that map to the identity in the codomain
• Always a normal subgroup—$\ker(f) \trianglelefteq G$, which is why kernels are central to forming quotient groups
• Measures injectivity—a homomorphism is injective (one-to-one) if and only if $\ker(f) = \{e_G\}$

Image of a Homomorphism

• Definition—the image $\text{im}(f) = \{f(g) \mid g \in G\}$ is the set of all elements in $H$ that are "hit" by the map
• Always a subgroup—$\text{im}(f) \leq H$, though not necessarily normal in $H$
• Measures surjectivity—a homomorphism is surjective (onto) if and only if $\text{im}(f) = H$

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Special Types: Isomorphisms and Automorphisms

When homomorphisms have additional properties—specifically bijectivity—they become powerful tools for proving that groups are structurally identical or for understanding a group's internal symmetries.

Isomorphisms

• Bijective homomorphism—an isomorphism is a homomorphism that is both injective ($\ker(f) = \{e\}$) and surjective ($\text{im}(f) = H$)
• Structural equivalence—if $G \cong H$ (groups are isomorphic), they have identical algebraic properties: same order, same subgroup structure, same everything
• The gold standard—proving two groups are isomorphic is often the goal of classification problems in group theory

Automorphisms

• Self-isomorphism—an automorphism is a bijective homomorphism from a group to itself, $f: G \to G$
• Internal symmetries—automorphisms represent the ways a group can be "rearranged" while preserving its structure
• Form their own group—the set of all automorphisms $\text{Aut}(G)$ forms a group under composition, revealing meta-structure

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The Big Theorem: First Isomorphism Theorem

This theorem is the crown jewel connecting homomorphisms, kernels, images, and quotient groups into one powerful statement. It tells you that every homomorphism factors through a quotient by its kernel.

The First Isomorphism Theorem

• The statement—if $f: G \to H$ is a homomorphism, then $G/\ker(f) \cong \text{im}(f)$
• Intuition—collapsing the kernel (what gets sent to identity) gives you a group that's isomorphic to what you actually hit in the codomain
• Central importance—this theorem appears constantly in proofs and provides the standard technique for constructing isomorphisms

Homomorphisms and Quotient Groups

• Kernel enables quotients—because $\ker(f)$ is always normal, you can always form the quotient group $G/\ker(f)$
• Coset structure—elements in the same coset of $\ker(f)$ all map to the same element in $H$, partitioning the domain by where elements land
• Analysis tool—quotient groups let you study "simplified" versions of groups by factoring out complexity

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Concrete Examples: Homomorphisms in Action

Abstract definitions become clear through examples. These canonical homomorphisms illustrate the range of behaviors possible and frequently appear on exams.

Examples of Homomorphisms in Group Theory

• Trivial homomorphism—maps every element to the identity $e_H$; kernel is all of $G$, image is just $\{e_H\}$
• Determinant map—$\det: GL_n(\mathbb{R}) \to \mathbb{R}^*$ is a homomorphism with kernel $SL_n(\mathbb{R})$ (matrices with determinant 1)
• Inclusion map—if $H \leq G$, the map $i: H \hookrightarrow G$ defined by $i(h) = h$ is an injective homomorphism

Relationship Between Homomorphisms and Subgroups

• Kernel is a subgroup—specifically a normal subgroup of the domain, which is essential for quotient construction
• Image is a subgroup—of the codomain, allowing you to identify important substructures
• Subgroup detection—homomorphisms provide a systematic way to find and classify subgroups through their kernels and images

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Quick Reference Table

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Self-Check Questions

1. If a homomorphism $f: G \to H$ has trivial kernel $\ker(f) = \{e_G\}$, what can you conclude about the map? What additional condition would make it an isomorphism?

2. Compare and contrast the kernel and image of a homomorphism: where does each live, what structure does each have, and what does each measure about the homomorphism's behavior?

3. Given the determinant map $\det: GL_n(\mathbb{R}) \to \mathbb{R}^*$, identify its kernel and use the First Isomorphism Theorem to describe the quotient $GL_n(\mathbb{R})/SL_n(\mathbb{R})$.

4. Why must the kernel of a homomorphism be a normal subgroup rather than just any subgroup? How does this connect to the construction of quotient groups?

5. Explain why two isomorphic groups must have the same order, the same number of elements of each order, and the same subgroup structure. What property of isomorphisms guarantees this?