Homomorphisms

Tom Judson · GFDL · Abstract Algebra: Theory and Applications, Ch. 3

A homomorphism is a function between groups that preserves the operation: f(a * b) = f(a) * f(b). The kernel measures what it forgets. The image measures what it hits. These three pieces tell you everything about how one group maps onto another.

Structure-preserving maps

A function f: G to H is a group homomorphism if f(a *_G b) = f(a) *_H f(b) for all a, b in G. The operation on the left is G's operation; the operation on the right is H's. The function translates one group's structure into the other's without distortion.

Scheme

; Homomorphism: f(a*b) = f(a)*f(b)
; Example: f: Z -> Z6, f(n) = n mod 6
; G = (Z, +), H = (Z6, + mod 6)

(define (f n) (modulo n 6))

; Check: f(a + b) = f(a) +6 f(b)
(define a 8)
(define b 15)

(display "f(a + b) = f(") (display (+ a b))
(display ") = ") (display (f (+ a b))) (newline)

(display "f(a) + f(b) mod 6 = ")
(display (modulo (+ (f a) (f b)) 6)) (newline)

; They match! The mod function is a homomorphism.
(display "Equal? ") (display (= (f (+ a b))
                                (modulo (+ (f a) (f b)) 6)))

Kernel

The kernel of f is the set of elements that map to the identity: ker(f) = (a in G : f(a) = e_H). The kernel is always a subgroup of G. A homomorphism is injective if and only if its kernel is trivial (just the identity).

Scheme

; Kernel of f: Z -> Z6, f(n) = n mod 6
; ker(f) = all integers n where n mod 6 = 0
; That is: ..., -12, -6, 0, 6, 12, ... = 6Z

(define (f n) (modulo n 6))

; Check a range
(display "Kernel elements in [-12..12]: ")
(let loop ((n -12))
  (when (<= n 12)
    (when (= (f n) 0)
      (display n) (display " "))
    (loop (+ n 1))))
(newline)

; The kernel is the subgroup 6Z of Z.
; It is isomorphic to Z itself.
(display "ker(f) = 6Z = multiples of 6")

Image

The image of f is the set of elements in H that are actually hit: im(f) = (f(a) : a in G). The image is always a subgroup of H. A homomorphism is surjective if its image is all of H.

Scheme

; Image of f: Z4 -> Z4, f(n) = 2n mod 4

(define (f n) (modulo (* 2 n) 4))

(display "f(0) = ") (display (f 0)) (newline)
(display "f(1) = ") (display (f 1)) (newline)
(display "f(2) = ") (display (f 2)) (newline)
(display "f(3) = ") (display (f 3)) (newline)

; Image = (0, 2). A subgroup of Z4.
; Not surjective (1 and 3 are not hit).
(display "Image = (0, 2)")

Notation reference

Math	Scheme	Python	Meaning
f: G → H	(define (f x) ...)	def f(x): ...	Homomorphism from G to H
ker(f)	(= (f x) 0)	f(x) == 0	Elements mapping to identity
im(f)	(map f G)	set(f(x) for x in G)	Elements in H that are hit
f(ab) = f(a)f(b)	verified above	verified above	Homomorphism property

Neighbors

Algebra track

← Ch.2 Subgroups — kernels and images are subgroups
Ch.4 Isomorphisms — bijective homomorphisms
🐱 Category Theory Ch.7 — functors are homomorphisms between categories
📐 Linear Algebra Ch.3 — linear maps are homomorphisms of vector spaces
✎ Proofs Ch.6 — the function properties (injective, surjective) used to classify homomorphisms

Category theory connections

Milewski Ch.7 — functors are homomorphisms between categories: they preserve composition and identity

Translation notes

Judson covers homomorphisms in Ch. 11 and isomorphisms in Ch. 9, after treating permutation groups and cosets. We pull homomorphisms forward because the concept is simpler than the textbook ordering suggests: it is just "f preserves the operation." The first isomorphism theorem (G/ker(f) is isomorphic to im(f)) is the punchline of this material but requires quotient groups, covered implicitly in Ch. 5.

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