Left Bruck loop

This article defines a property that can be evaluated for a loop.
View other properties of loops

Definition

Bruck-Bol definition

A loop is said to be a left Bruck loop or left K-loop or left gyrocommutative gyrogroup if it satisfies the following conditions:

• It is a left Bol loop
• It satisfies the automorphic inverse property (note that on account of its being a left Bol loop, it is already an inverse loop)

Gyrogroup definition

A gyrogroup $(G,*)$ with gyration map $\operatorname {gyr}$ is termed a left Bruck loop or left K-loop or left gyrocommutative gyrogroup if it satisfies the following identity called gyrocommutativity:

$a*b=\operatorname {gyr} [a,b](b*a)$

Relation with other properties

Weaker properties

• Automorphic inverse property
• Left Bol loop
• Inverse loop
• Power-associative loop

References

• Finite Bruck loops by Michael Aschbacher, Michael K. Kinyon, and J. D. Phillips

External links

• Arxiv link for Finite Bruck loops by Aschbacher, Kinyon and Phillips