Left gyrogroup

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Definition

A magma with underlying set G and binary operation * is termed a gyrogroup if the following hold:

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

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