Pseudoautomorphisms of Bruck loops and their generalizations

Mark Greer; Michael Kinyon

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 3, page 383-389
  • ISSN: 0010-2628

Abstract

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We show that in a weak commutative inverse property loop, such as a Bruck loop, if α 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.

How to cite

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Greer, Mark, and Kinyon, Michael. "Pseudoautomorphisms of Bruck loops and their generalizations." Commentationes Mathematicae Universitatis Carolinae 53.3 (2012): 383-389. <http://eudml.org/doc/246385>.

@article{Greer2012,
abstract = {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.},
author = {Greer, Mark, Kinyon, Michael},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {pseudoautomorphism; Bruck loop; weak commutative inverse property; pseudoautomorphisms; Bruck loops; weak commutative inverse property loops},
language = {eng},
number = {3},
pages = {383-389},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Pseudoautomorphisms of Bruck loops and their generalizations},
url = {http://eudml.org/doc/246385},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Greer, Mark
AU - Kinyon, Michael
TI - Pseudoautomorphisms of Bruck loops and their generalizations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 3
SP - 383
EP - 389
AB - 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.
LA - eng
KW - pseudoautomorphism; Bruck loop; weak commutative inverse property; pseudoautomorphisms; Bruck loops; weak commutative inverse property loops
UR - http://eudml.org/doc/246385
ER -

References

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