Group: associative, with an identity and an inverse for every element¶

Level: 101 · one row of the table on the laws of an operation, on its own page

One line: A group is a monoid in which every element can be undone, so every equation a · x = b has exactly one solution x = a⁻¹ · b; ℤ under +, nonzero fractions under ×, and the six shufflings of three objects are groups, and ℕ under + is not, because nothing undoes 3.

Laws required¶

associative, identity, inverses, of the four on the parent page.

What it is, and the examples¶

A group is a monoid in which every x has an inverse, a y with x · y = e = y · x. Simovici and Djeraba's Definition 1.143 writes it as an algebra of type (0, 2, 1): the constant e, the operation, and the unary operation x ↦ x⁻¹. The inverse is unique, by the one-line argument y = y · e = y · (x · y′) = (y · x) · y′ = e · y′ = y′.

The examples in the program: ℤ under + with inverse −n; the nonzero fractions under × with inverse 1/q; and the six permutations of three objects under composition, the smallest group that is not commutative, which the program shows by finding two shuffles that do not commute. The non-examples are ℕ under +, a monoid in which 3 has no inverse, and ℤ under ×, in which 2 has none: an inverse must exist inside the set, which is why "which elements can be divided by" is the question multiplication can be undone asks.

What inverses buy is solving: in a group, a · x = b has exactly one solution, x = a⁻¹ · b, so every element appears exactly once in every row and column of the operation table, a Latin square. That is the structure behind cancelling in equations, behind every symmetry (the symmetries of any object form a group under composition), and behind the arithmetic of ℤ/mℤ that equivalence relations builds.

What the program prints¶

Verified output of structure_group.py — regenerated by tools/run_examples.py, never hand-typed.

LAWS REQUIRED: associative, identity, inverses
   integers mod 7 under + (inverse is 7 − n)
     associative: True
     identity:    True   it is 0
     inverses:    True
     commutative: True
   nonzero fractions (sample) under ×
     associative: True
     identity:    True   it is 1
     inverses:    True
     commutative: True
   the 6 permutations of 3 objects under composition
     associative: True
     identity:    True   it is (0, 1, 2)
     inverses:    True
     commutative: False   fails at (0, 2, 1), (1, 0, 2)
   ℕ 0..6 under + (the non-example)
     associative: True
     identity:    True   it is 0
     inverses:    False   none for 1, 2, 3
     commutative: True
   in a group every row of the table is a rearrangement of the set (a Latin square):
     (0, 1, 2) ∘ _ : True
     (0, 2, 1) ∘ _ : True
     (1, 0, 2) ∘ _ : True

Po polsku, w skrócie¶

Grupa to monoid, w którym każdy element ma odwrotność: x · y = e = y · x. Przykłady: ℤ z + (odwrotność −n), ułamki niezerowe z × (odwrotność 1/q), sześć permutacji trzech obiektów ze składaniem, najmniejsza grupa nieprzemienna. Kontrprzykłady: ℕ z + (3 nie ma odwrotności), ℤ z × (2 nie ma). Odwrotności dają rozwiązywanie równań: a · x = b ma dokładnie jedno rozwiązanie x = a⁻¹ · b, a tabela działania jest kwadratem łacińskim.

Auf Deutsch: Stichwörter¶

Gruppe: assoziativ, mit neutralem Element und einem Inversen für jedes Element.

Stichwörter: Gruppe, neutrales Element, Inverses, Assoziativgesetz, Gruppentafel, lateinisches Quadrat.