Pseudoautomorphisms of Bruck loops and their generalizations

TL;DR

This paper investigates pseudoautomorphisms in Bruck loops and their generalizations, establishing conditions under which these mappings are automorphisms, especially in loops with trivial nuclei or commutative inverse properties.

Contribution

It generalizes Bruck's result by showing pseudoautomorphisms are automorphisms in broader classes of loops, including weak commutative inverse property loops.

Findings

Pseudoautomorphisms with certain companions lie in the left nucleus.

In loops with trivial left nucleus, all pseudoautomorphisms are automorphisms.

In commutative inverse property loops, all pseudoautomorphisms are automorphisms.

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.