# Gyrogroup

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QUICK PHRASES: left identity, left inverses and associativity twisted by an automorphism

## Definition

### Minimal definition

A magma with underlying set $G$ and binary operation $*$ is termed a gyrogroup if the following hold:

• Left identity and left inverse: There is an element $e \in G$ such that $e$ is a left neutral element and every element has a left inverse with respect to $e$. In other words:

$e * a = a \forall a \in G$

and for all $a \in G$, there exists $b \in G$ such that:

$b * a = e$

• Gyroassociativity: For any $a,b,c \in G$, there is a unique element $\operatorname{gyr}[a,b]c \in G$ such that:

$a * (b * c) = (a * b) * (\operatorname{gyr}[a,b]c)$

• Gyroautomorphism: $\operatorname{gyr}[a,b]$ (i.e., the map that sends $c$ to $\operatorname{gyr}[a,b]c$) is a magma automorphism of $G$. This is called the Thomas gyration, or gyroautomorphism, of $G$.
• Left loop property: The following are equal as automorphisms of $G$:

$\operatorname{gyr}[a,b] = \operatorname{gyr}[a * b,b]$

### Maximal definition

A magma with underlying set $G$ and binary operation $*$ is termed a gyrogroup if the following hold:

• Two-sided identity and two-sided inverse: There is a unique element $e \in G$ such that $e$ is a two-sided neutral element and every element has a unique two-sided inverse element with respect to $e$. In other words:

$e * a = a * e = a\forall a \in G$

and for all $a \in G$, there exists a unique two-sided inverse $b \in G$ such that:

$b * a = a * b = e$

The element $b$ is denoted $a^{-1}$.

• Gyroassociativity: For any $a,b,c \in G$, there is a unique element $\operatorname{gyr}[a,b]c \in G$ such that:

$a * (b * c) = (a * b) * (\operatorname{gyr}[a,b]c)$

• Gyroautomorphism: $\operatorname{gyr}[a,b]$ (i.e., the map that sends $c$ to $\operatorname{gyr}[a,b]c$) is a magma automorphism of $G$. This is called the Thomas gyration, or gyroautomorphism, of $G$.
• Left loop property: The following are equal as automorphisms of $G$:

$\operatorname{gyr}[a,b] = \operatorname{gyr}[a * b,b]$

### Equivalence of definitions

Further information: equivalence of definitions of gyrogroup

## Facts

### Embeddings inside groups

Gyrogroups are closely related to twisted subgroups as follows: Every gyrogroup can be embedded as a twisted subgroup of some group. In general, a twisted subgroup need not be a gyrogroup.

## References

• Involutory decomposition of groups into twisted subgroups and subgroups by Tuval Foguel and Abraham A. Ungar