Homomorphisms and Quotient Groups - Core Definitions

Homomorphisms are structure-preserving maps between groups, providing the natural notion of morphism in group theory. They reveal how groups relate to one another and enable the construction of quotient groups.

DefinitionGroup Homomorphism

A function $\phi: G \to H$ between groups is a homomorphism if it preserves the group operation: $\phi(ab) = \phi(a)\phi(b) \quad \text{for all } a, b \in G$

Key properties of homomorphisms:

$\phi(e_G) = e_H$ (identity maps to identity)
$\phi(a^{-1}) = \phi(a)^{-1}$ (inverses map to inverses)
$\phi(a^n) = \phi(a)^n$ for all $n \in \mathbb{Z}$

ExampleClassical Homomorphisms

Determinant: $\det: GL_n(\mathbb{R}) \to \mathbb{R}^*$ satisfies $\det(AB) = \det(A)\det(B)$
Exponential: $\exp: (\mathbb{R}, +) \to (\mathbb{R}^+, \cdot)$ satisfies $\exp(a+b) = \exp(a)\exp(b)$
Sign map: $\text{sgn}: S_n \to \{\pm 1\}$ maps even permutations to $+1$ and odd to $-1$
Modular reduction: $\pi: \mathbb{Z} \to \mathbb{Z}_n$ given by $\pi(a) = a \bmod n$

DefinitionKernel and Image

For a homomorphism $\phi: G \to H$ :

The kernel is: $\ker(\phi) = \{g \in G : \phi(g) = e_H\}$

The image is: $\text{im}(\phi) = \{h \in H : h = \phi(g) \text{ for some } g \in G\}$

The kernel is always a normal subgroup of $G$ , and the image is always a subgroup of $H$ .

The kernel measures how far $\phi$ is from being injective. In fact, $\phi$ is injective (one-to-one) if and only if $\ker(\phi) = \{e_G\}$ . An injective homomorphism is called a monomorphism, a surjective homomorphism is an epimorphism, and a bijective homomorphism is an isomorphism.

Remark

Two groups $G$ and $H$ are isomorphic (written $G \cong H$ ) if there exists an isomorphism between them. Isomorphic groups have identical algebraic structure—they are essentially the same group with different labels. For example, $\mathbb{Z}_4$ and the group of rotations of a square are isomorphic.

Understanding homomorphisms is crucial because they allow us to study groups through their relationships. The First Isomorphism Theorem establishes that every homomorphism induces an isomorphism $G/\ker(\phi) \cong \text{im}(\phi)$ , connecting quotient groups to subgroups of the target.