# nLab nonassociative group

Contents

### Context

#### Algebra

higher algebra

universal algebra

# 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 $(G,\backslash,/,1)$ with a unary operation called the inverse $(-)^{-1}:G \to G$ such that

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

for all $a \in G$.

### Without division

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

• $a^{-1} \cdot a = 1$
• $a \cdot a^{-1} = 1$
• $(a \cdot b^{-1}) \cdot b = a$
• $(a \cdot b) \cdot b^{-1} = a$
• $b \cdot (b^{-1} \cdot a) = a$
• $b^{-1} \cdot (b \cdot a) = a$

for all $a,b \in G$.

## 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.