# Left gyrogroup

## 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:

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

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 $gyr[a,b]c \in G$ such that:

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

• Gyroautomorphism: $\operatorname{gyr}[a,b]$ is a magma automorphism of $G$ for all $a,b \in G$. This is called the Thomas gyration, or gyroautomorphism, of $G$.
• Weak loop property: If $b * a = e$, then

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

Thus, the definition of left gyrogroup differs from that of gyrogroup, only in that the left loop property is relaced by the weak loop property.

## Facts

### Representations of left gyrogroups

By a representation of a left gyrogroup, we mean an expression of it as a left gyrotransversal of a subgroup.

## References

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