Isomorphisms

Tom Judson · GFDL · Abstract Algebra: Theory and Applications, Ch. 4

An isomorphism is a bijective homomorphism. When two groups are isomorphic, they have the same structure wearing different labels. Cayley's theorem says every group is isomorphic to a group of permutations, so permutation groups are all there is.

When two groups are the same

Groups G and H are isomorphic (written G ≅ H) if there exists a bijective homomorphism f: G → H. Bijective means one-to-one and onto. Isomorphic groups have the same Cayley table up to relabeling.

Scheme

; Z2 = {0, 1} under + mod 2
; (1, -1) under multiplication
; They are isomorphic: f(0)=1, f(1)=-1

(define (z2+ a b) (modulo (+ a b) 2))
(define (signs* a b) (* a b))

; Map: 0 -> 1, 1 -> -1
(define (f x) (if (= x 0) 1 -1))

; Check homomorphism: f(a+b) = f(a) * f(b)
(display "f(0+0) = ") (display (f (z2+ 0 0)))
(display "  f(0)*f(0) = ") (display (signs* (f 0) (f 0))) (newline)

(display "f(0+1) = ") (display (f (z2+ 0 1)))
(display "  f(0)*f(1) = ") (display (signs* (f 0) (f 1))) (newline)

(display "f(1+0) = ") (display (f (z2+ 1 0)))
(display "  f(1)*f(0) = ") (display (signs* (f 1) (f 0))) (newline)

(display "f(1+1) = ") (display (f (z2+ 1 1)))
(display "  f(1)*f(1) = ") (display (signs* (f 1) (f 1)))
; All match. f is a homomorphism, and it is bijective.

Cayley's theorem

Every group G is isomorphic to a subgroup of the symmetric group on G's elements (Cayley's theorem). The map sends each element g to the permutation "multiply everything by g on the left." This means every abstract group is secretly a group of permutations.

Scheme

; Cayley's theorem for Z3
; Each element g maps to the permutation: x -> g+x mod 3

(define (cayley-perm g n)
  (let loop ((x 0) (result '()))
    (if (= x n)
        (reverse result)
        (loop (+ x 1) (cons (modulo (+ g x) n) result)))))

(display "0 -> ") (display (cayley-perm 0 3)) (newline)
(display "1 -> ") (display (cayley-perm 1 3)) (newline)
(display "2 -> ") (display (cayley-perm 2 3)) (newline)
; Each element becomes a permutation of (0,1,2).
; These three permutations form a subgroup of S3.

Notation reference

Math	Scheme	Python	Meaning
G ≅ H	bijective f	bijective f	G and H are isomorphic
S_n	permutation lists	permutation lists	Symmetric group on n elements
L_g(x) = gx	(cayley-perm g n)	left multiplication	Cayley embedding

Neighbors

Algebra track

← Ch.3 Homomorphisms — isomorphisms are bijective homomorphisms
Ch.5 Cosets and Lagrange's Theorem — subgroup structure constrains isomorphism classes

Category theory connections

Milewski Ch.7 — isomorphisms in a category are invertible morphisms; Cayley's theorem is a special case of the Yoneda lemma
🍞 Milewski Ch.1 — isomorphisms as the fundamental notion of "sameness" in categories

Translation notes

Judson covers isomorphisms in Ch. 9 after cosets and normal subgroups. We bring it forward because the concept requires only bijective homomorphisms. Cayley's theorem appears in Ch. 5 of Judson. The connection to the Yoneda lemma is not in Judson but is the categorical generalization: every object is determined by the morphisms into (or out of) it.

← Homomorphisms by june.kim Cosets and Lagrange's Theorem · 5 of 8 →