Gyrogroup
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QUICK PHRASES: left identity, left inverses and associativity twisted by an automorphism
Contents
Definition
Minimal definition
A magma with underlying set and binary operation is termed a gyrogroup if the following hold:
- Left identity and left inverse: There is an element such that is a left neutral element and every element has a left inverse with respect to . In other words:
and for all , there exists such that:
- Gyroassociativity: For any , there is a unique element such that:
- Gyroautomorphism: (i.e., the map that sends to ) is a magma automorphism of . This is called the Thomas gyration, or gyroautomorphism, of .
- Left loop property: The following are equal as automorphisms of :
Maximal definition
A magma with underlying set and binary operation is termed a gyrogroup if the following hold:
- Two-sided identity and two-sided inverse: There is a unique element such that is a two-sided neutral element and every element has a unique two-sided inverse element with respect to . In other words:
and for all , there exists a unique two-sided inverse such that:
The element is denoted .
- Gyroassociativity: For any , there is a unique element such that:
- Gyroautomorphism: (i.e., the map that sends to ) is a magma automorphism of . This is called the Thomas gyration, or gyroautomorphism, of .
- Left loop property: The following are equal as automorphisms of :
Equivalence of definitions
Further information: equivalence of definitions of gyrogroup
Relation with other structures
Stronger structures
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