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.

Relation with other structures

Stronger structures

• Group
• Gyrogroup

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

External links

• Weblink for Involutory decomposition of groups into twisted subgroups and subgroups by Foguel and Ungar