nLab
nonassociative group

Contents

Contents

Idea

A non-associative group, or an invertible loop. Nonassociative is used in the sense of not-necessarily associative, in the same sense that a nonassociative algebra is not-necessarily associative.

Definition

A nonassociative group or invertible loop is a loop (G,\,/,1)(G,\backslash,/,1) with a unary operation called the inverse () 1:GG(-)^{-1}:G \to G such that

  • a 1a=1a^{-1} \cdot a = 1
  • aa 1=1a \cdot a^{-1} = 1

for all aGa \in G.

Without division

A nonassociative group or invertible loop is a unital magma (G,()():G×GG),1:G)(G,(-)\cdot(-):G\times G\to G),1:G) with a unary operation called the inverse () 1:GG(-)^{-1}:G \to G such that

  • a 1a=1a^{-1} \cdot a = 1
  • aa 1=1a \cdot a^{-1} = 1
  • (ab 1)b=a(a \cdot b^{-1}) \cdot b = a
  • (ab)b 1=a(a \cdot b) \cdot b^{-1} = a
  • b(b 1a)=ab \cdot (b^{-1} \cdot a) = a
  • b 1(ba)=ab^{-1} \cdot (b \cdot a) = a

for all a,bGa,b \in G.

Properties

Every non-associative group is a loop with a two-sided inverse.

Examples

  • Every group is a nonassociative group.

Last revised on May 25, 2021 at 10:35:40. See the history of this page for a list of all contributions to it.