Group Homomorphisms — Mathematics Knowledge System

Overview

A group homomorphism is a structure-preserving map between two groups: it sends products to products, so the algebraic relationships in the source group are mirrored exactly in the target group. Homomorphisms let us compare groups, transfer theorems between them, and formally define when two groups are 'the same' (isomorphic).

Intuition

Imagine translating sentences between two languages that both encode the same underlying logic — as long as the translation preserves how sentences combine (concatenating two source sentences corresponds to concatenating their translations), the translation is faithful to the algebraic structure. A group homomorphism is exactly this kind of faithful translation between two groups: it doesn't need to be a perfect one-to-one dictionary (that stronger property is an isomorphism), but it must respect the operation.

Formal Definition

Definition

Let (G, ·) and (H, ∗) be groups. A function φ: G → H is a group homomorphism if it preserves the group operation:

\varphi(a \cdot b) = \varphi(a) * \varphi(b) \quad \forall a, b \in G

Homomorphism property

\ker(\varphi) = \{ g \in G : \varphi(g) = e_H \}

Kernel of φ

\operatorname{im}(\varphi) = \{ \varphi(g) : g \in G \}

Image of φ

Notation

Notation	Meaning
$\varphi: G \to H$	A homomorphism from G to H
$\ker(\varphi)$	Kernel — elements mapped to the identity of H
$G \cong H$	G is isomorphic to H (a bijective homomorphism exists)
$G/N$	Quotient group of G by a normal subgroup N

Properties

Identity and inverses are preserved

\varphi(e_G) = e_H \quad \text{and} \quad \varphi(a^{-1}) = \varphi(a)^{-1}.

Kernel is a normal subgroup

\ker(\varphi) \trianglelefteq G.

Image is a subgroup

\operatorname{im}(\varphi) \leq H.

Injectivity criterion

\varphi \text{ is injective} \iff \ker(\varphi) = \{e_G\}.

Applications

Hash functions used in some cryptographic constructions are designed to approximate homomorphic behavior (e.g. homomorphic encryption preserves operations on encrypted data).

Worked Examples

Check the homomorphism property directly.
$\varphi(a+b) = (a+b) \bmod n = (a \bmod n + b \bmod n) \bmod n = \varphi(a) + \varphi(b)$
The kernel is all integers mapping to 0 mod n, i.e. all multiples of n.
$\ker(\varphi) = n\mathbb{Z}$

Answer: φ is a homomorphism (the 'reduction mod n' map), and ker(φ) = nℤ, giving the isomorphism ℤ/nℤ ≅ ℤₙ.

Practice Problems

Difficulty 6/10

Is the map φ: (ℝ, +) → (ℝ\{0}, ×) given by φ(x) = eˣ a homomorphism? What are its kernel and image?

Common Mistakes

Common Mistake

Assuming any function between groups that happens to be bijective is automatically a homomorphism.

Bijectivity says nothing about preserving the operation; both properties (structure-preserving AND bijective) are needed for an isomorphism.

Common Mistake

Confusing the kernel with the identity element itself.

The kernel is a whole subgroup (possibly with many elements) that maps to the identity of the codomain — not just the identity of the domain.

Summary

A homomorphism φ: G → H satisfies φ(ab) = φ(a)φ(b), preserving the group operation.
The kernel ker(φ) is always a normal subgroup of G; the image is always a subgroup of H.
φ is injective exactly when its kernel is trivial: ker(φ) = {e}.
A bijective homomorphism is an isomorphism, meaning the two groups have identical structure (G ≅ H).
The first isomorphism theorem states G/ker(φ) ≅ im(φ).

References

WebsiteWikipedia — Group homomorphism
BookDummit, D. & Foote, R. Abstract Algebra, 3rd ed., Ch. 3.