# Homomorphism & Isomorphism

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/homomorphism-isomorphism/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A homomorphism preserves algebraic structure between objects. An isomorphism is a perfectly reversible homomorphism, showing two structures are identical.

## Definition
φ:G→H is a homomorphism if φ(a★b)=φ(a)∘φ(b) for all a,b∈G.

## Kernel and image
ker(φ)={g:φ(g)=eH} — always a normal subgroup. im(φ) — always a subgroup of H.

## First Isomorphism Theorem
G/ker(φ) ≅ im(φ)

## Examples
Rotations of a square ≅ ℤ/4ℤ. Positive reals under × ≅ reals under + (via log). Any two groups of prime order p are isomorphic.

## Field homomorphisms are automatically injective
Since a field's only ideals are {0} and the whole field.

## Category theory
Homomorphisms are the prototypical morphisms (Eilenberg-Mac Lane, 1945).

## Sources
### Tier 1
- Dummit, D.S. and Foote, R.M. (2004). *Abstract Algebra*. 3rd ed.
- Lang, S. (2002). *Algebra*. Revised 3rd ed.
### Tier 2
- Mac Lane, S. (1998). *Categories for the Working Mathematician*. 2nd ed.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*