nLab nonassociative group

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.

Related concepts

• group (associative version)

• loop (non-invertible version)

• invertible quasigroup (non-unital version)

• commutative loop (commutative version)

• centipede mathematics