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:

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

Weaker structures

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