Engineering / Mathematics — Abelian Groups
Group Homomorphisms and Isomorphisms
Structure-preserving maps between groups, isomorphisms, and what it means for two groups to be the same.
Executive summary
A homomorphism preserves the group operation. An isomorphism is a bijective homomorphism, and isomorphic groups are indistinguishable as abstract structures.
The gap between abstract sameness and computational sameness is where public-key cryptography lives.
Learning objectives
- Define homomorphisms and derive their basic properties.
- Distinguish isomorphism from equality.
- Explain why isomorphic groups can differ computationally.
01Homomorphisms
Group homomorphism
A map f : G → G' with f(ab) = f(a)f(b) for all a, b ∈ G.
Two consequences follow immediately and require no extra hypotheses: f(e) = e', and f(a⁻¹) = f(a)⁻¹. Both are proved by applying the defining property and cancelling.
| Map | Domain → codomain | Homomorphism? |
|---|---|---|
| a ↦ a^k | G → G, G abelian | Yes |
| a ↦ a mod n | Z → Z_n | Yes |
| Legendre symbol | Z_p* → {±1} | Yes |
| a ↦ a + 1 | Z → Z | No; fails at the identity |
| Frobenius x ↦ x^p | F_q → F_q | Yes, for both operations |
02Isomorphisms
Isomorphism
A bijective homomorphism. Groups G and G' are isomorphic, written G ≅ G', if one exists.
Isomorphic groups have identical abstract structure: same order, same subgroup lattice, same element order distribution. Any statement expressible in group-theoretic terms holds for one exactly when it holds for the other.
Classification of cyclic groups
Every cyclic group of order n is isomorphic to Z_n under addition, and every infinite cyclic group is isomorphic to Z.
03Abstract sameness is not computational sameness
The resolution is that an isomorphism need not be efficiently computable. The map x ↦ g^x from Z_q to the subgroup is an isomorphism, easy to evaluate and believed hard to invert. Computing the inverse isomorphism is the discrete logarithm problem.
This gap is the entire basis of discrete-log cryptography. The security does not come from the abstract group, which is as simple as a group can be, but from the difficulty of translating between two representations of it.
04Frequently asked questions
Is every bijective homomorphism an isomorphism?
For groups, yes — the inverse of a bijective homomorphism is automatically a homomorphism. This is a convenience specific to algebraic structures of this kind and fails in some other categories.
How can two groups be shown non-isomorphic?
By exhibiting a structural difference: unequal orders, different numbers of elements of a given order, or different subgroup lattices. Z_4 and Z_2 × Z_2 both have order 4 but differ in whether an element of order 4 exists.
Does an efficiently computable isomorphism always exist between isomorphic groups?
No, and the discrete logarithm problem is the standard counterexample. Existence of an isomorphism is a statement about abstract structure and carries no computational content.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 194-202.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.