Groups

Tom Judson · GFDL · Abstract Algebra: Theory and Applications, Ch. 1

A group is a set with one operation that satisfies four axioms: closure, associativity, identity, and inverses. It is the simplest algebraic structure that captures symmetry. Every symmetry you will ever encounter is a group.

The four axioms

A set G with a binary operation * is a group if: (1) for all a, b in G, a * b is in G (closure); (2) for all a, b, c in G, (a * b) * c = a * (b * c) (associativity); (3) there exists an element e in G such that e * a = a * e = a (identity); (4) for every a in G, there exists a⁻¹ such that a * a⁻¹ = a⁻¹ * a = e (inverse).

Scheme

; The integers under addition form a group.
; Closure: sum of two integers is an integer.
; Associativity: (a + b) + c = a + (b + c).
; Identity: 0.
; Inverse: -a.

(define (op a b) (+ a b))
(define identity 0)
(define (inverse a) (- a))

; Check associativity
(define a 3) (define b 5) (define c 7)
(display "(a+b)+c = ") (display (op (op a b) c)) (newline)
(display "a+(b+c) = ") (display (op a (op b c))) (newline)

; Check identity
(display "a+0 = ") (display (op a identity)) (newline)

; Check inverse
(display "a+(-a) = ") (display (op a (inverse a)))

Symmetries of a triangle

The symmetries of an equilateral triangle form a group with six elements: three rotations (0, 120, 240 degrees) and three reflections. This is the symmetric group S3, the smallest non-abelian group. Composing two symmetries gives another symmetry.

Scheme

; Permutations as lists: (1 2 3) means 1->1, 2->2, 3->3
; We represent a permutation as a function on indices 1,2,3.

(define (apply-perm p i) (list-ref p (- i 1)))

; S3 elements
(define e  '(1 2 3))  ; identity
(define r1 '(3 1 2))  ; rotate 120
(define r2 '(2 3 1))  ; rotate 240
(define s1 '(1 3 2))  ; reflect (fix 1)
(define s2 '(3 2 1))  ; reflect (fix 2)
(define s3 '(2 1 3))  ; reflect (fix 3)

; Compose two permutations
(define (compose-perm p q)
  (list (apply-perm p (apply-perm q 1))
        (apply-perm p (apply-perm q 2))
        (apply-perm p (apply-perm q 3))))

(display "r1 then r1 = ") (display (compose-perm r1 r1)) (newline)
; rotate 120 twice = rotate 240
(display "r2         = ") (display r2) (newline)

(display "s1 then s1 = ") (display (compose-perm s1 s1)) (newline)
; reflect twice = identity

(display "r1 then s1 = ") (display (compose-perm r1 s1)) (newline)
(display "s1 then r1 = ") (display (compose-perm s1 r1))
; Different! S3 is non-abelian.

Cayley tables

A Cayley table records every product in a finite group. Each row and each column is a permutation of the group elements (this is a consequence of the group axioms). Reading the table tells you everything about the group.

Scheme

; Cayley table for Z4 = integers mod 4 under addition
; Elements: 0, 1, 2, 3

(define (mod4+ a b) (modulo (+ a b) 4))

(display "+  | 0  1  2  3") (newline)
(display "---+------------") (newline)
(let loop ((i 0))
  (when (< i 4)
    (display i) (display "  | ")
    (let inner ((j 0))
      (when (< j 4)
        (display (mod4+ i j)) (display "  ")
        (inner (+ j 1))))
    (newline)
    (loop (+ i 1))))
; Every row is a permutation of 0,1,2,3.

Notation reference

Math	Scheme	Python	Meaning
a * b	(op a b)	op(a, b)	Group operation
e	identity	identity	Identity element
a⁻¹	(inverse a)	inverse(a)	Inverse of a
S₃	'(3 1 2)	[3,1,2]	Permutation
Z₄	(modulo (+ a b) 4)	(a+b)%4	Integers mod 4

Neighbors

Algebra track

Ch.2 Subgroups — cyclic groups and subgroup tests

Category theory connections

Milewski Ch.3 — monoids are groups without inverses; a group is a one-object category where every morphism is invertible

Translation notes

Judson's textbook opens with preliminaries on sets and equivalence relations, which we skip. The treatment here covers chapters 3 (Groups) and 4 (Cyclic Groups) of the textbook, compressing the core definitions into executable examples. S3 is built from list permutations, which is less elegant than Judson's cycle notation but easier to run in Scheme. All groups here are finite.

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