Engineering / Mathematics — Abelian Groups
Kernels, Images and the Isomorphism Theorems
The kernel and image of a homomorphism, and the first isomorphism theorem relating them to a quotient.
Executive summary
The kernel of a homomorphism measures how far it is from injective, and the image measures how far from surjective. The first isomorphism theorem states the exact relationship between the two.
The theorem is the standard tool for identifying a quotient group with something concrete.
Learning objectives
- Define kernel and image and verify they are subgroups.
- State and apply the first isomorphism theorem.
- Use kernels to test injectivity.
01Kernel and image
Kernel and image
ker f = {a ∈ G : f(a) = e'} and im f = {f(a) : a ∈ G}.
Both are subgroups
ker f is a subgroup of G and im f is a subgroup of G'.
Moreover f is injective if and only if ker f = {e}.
The injectivity criterion is the practical value of the kernel. Rather than comparing all pairs, one checks a single subgroup for triviality — f(a) = f(b) is equivalent to f(ab⁻¹) = e', so collisions correspond exactly to non-identity kernel elements.
02The first isomorphism theorem
First isomorphism theorem
For a homomorphism f : G → G',
G / ker f ≅ im f.
The isomorphism sends the coset a · ker f to f(a). Well-definedness is exactly the statement that elements of the same coset have the same image, and injectivity is that different cosets have different images.
Take a homomorphism
Any structure-preserving map f from G.
Collapse the kernel
Form the quotient G / ker f, identifying elements with the same image.
Recover the image
The quotient is isomorphic to im f.
Count
|G| = |ker f| · |im f| for finite G.
03Applications
| Homomorphism | Kernel | Image | Theorem gives |
|---|---|---|---|
| Z → Z_n, reduction | nZ | Z_n | Z/nZ ≅ Z_n |
| Z_p* → {±1}, Legendre | Quadratic residues | {±1} | Index of residues is 2 |
| G → G, a ↦ a^k | Elements of order dividing k | k-th powers | Counts k-th powers |
| Z_{mn}* → Z_m* × Z_n* | Trivial for coprime m, n | Everything | Chinese remainder theorem |
The second row settles a fact used throughout the quadratic residue stream: exactly half the non-zero residues modulo an odd prime are squares. The kernel of the Legendre map has index 2 by the theorem, and the image has two elements, so the count follows without any separate argument.
04Frequently asked questions
Why is the kernel automatically a normal subgroup?
Because conjugating a kernel element leaves it in the kernel, as f respects the operation. In the abelian setting this is vacuous since all subgroups are normal, but it is what makes the theorem work in general.
Does the theorem require finiteness?
No, it holds for arbitrary groups. Only the counting corollary |G| = |ker f| · |im f| requires finite order.
How is it used in practice here?
Chiefly to count. Establishing that a map is a homomorphism and identifying its kernel immediately gives the size of the image, which is how the number of quadratic residues and the number of k-th powers in a cyclic group are determined.
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.