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

Relation with other structures

Stronger structures

• Group

Weaker structures

• Left gyrogroup
• Left-inverse property loop: For full proof, refer: gyrogroup implies left-inverse property loop

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

External links

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