Pseudoautomorphisms of Bruck loops and their generalizations

Abstract

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.A loop (Q, •) consists of a set Q with a binary operation • : Q × Q → Q such that (i) for all a, b ∈ Q, the equations ax = b and ya = b have unique solutions x, y ∈ Q, and (ii) there exists 1 ∈ Q such that 1x = x1 = x for all x ∈ Q. We denote these unique solutions by x = a\b and y = b/a, respectively. For x ∈ Q, define the right and left translations by x by, respectively, yR x = yx and yL x = xy for all y ∈ Q. That these mappings are permutations of Q is essentially part of the definition of loop. Standard reference in loop theory are [7,13].A triple (α, β, γ) of permutations of a loop Q is an autotopism if for all x, y ∈ Q, xα • yβ = (xy)γ. The set Atp(Q) of all autotopisms of Q is a group under composition. Of particular interest here are the three subgroupsand so every element of Atp λ (Q) has the form (βL a , β, βL a ) for some a ∈ Q. Conversely, it is easy to see that if a triple of permutations of that form is an autotopism, then 1β = 1.By similar arguments for the other two cases, we have the following characterizations:Since these special types of autotopisms are entirely determined by a single permutation and an element of the loop, it is customary to focus on those instead of on the autotopisms themselves. This motivates the following definitions.Let Q be a loop. If β ∈ Sym(Q) and a ∈ Q satisfy a • (xy)β = (a • xβ)(yβ) (1) 2010 Mathematics Subject Classification. 20N05.