Geodesic
Pseudoautomorphisms of Bruck loops and their generalizations
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 3, pp. 383-389

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We show that in a weak commutative inverse property loop, such as a Bruck loop, if $\alpha$ is a right [left] pseudoautomorphism with companion $c$, then $c$ [$c^2$] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism, generalizing a well-known result of Bruck.
We show that in a weak commutative inverse property loop, such as a Bruck loop, if $\alpha$ is a right [left] pseudoautomorphism with companion $c$, then $c$ [$c^2$] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism, generalizing a well-known result of Bruck.
Classification : 20N05
Keywords: pseudoautomorphism; Bruck loop; weak commutative inverse property 
Greer, Mark; Kinyon, Michael. Pseudoautomorphisms of Bruck loops and their generalizations. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 3, pp. 383-389. http://geodesic.mathdoc.fr/item/CMUC_2012_53_3_a4/
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     author = {Greer, Mark and Kinyon, Michael},
     title = {Pseudoautomorphisms of {Bruck} loops and their generalizations},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {383--389},
     year = {2012},
     volume = {53},
     number = {3},
     mrnumber = {3017837},
     zbl = {1256.20062},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2012_53_3_a4/}
}
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