Gyrogroup

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 $$G$$:

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

Stronger structures

 * Weaker than::Group

Weaker structures

 * Stronger than::Left gyrogroup
 * Stronger than::Left-inverse property loop:

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.