Gyrocommutative gyrogroup

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Definition

A gyrogroup G with gyromultiplication * and gyroautomorphism \operatorname{gyr} is defined to be gyrocommutative if:

a * b = \operatorname{gyr}([a,b])(b * a) \ \forall \ a,b \in G

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
gyrogroup |FULL LIST, MORE INFO
left gyrogroup |FULL LIST, MORE INFO
left-inverse property loop |FULL LIST, MORE INFO