Pseudoautomorphisms of Bruck Loops and Their Generalizations

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] 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, yRx = yx and yLx = 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 subgroups Atpλ(Q) = {(α, β, γ) ∈ Atp(Q) | 1β = 1} , Atpμ(Q) = {(α, β, γ) ∈ Atp(Q) | 1γ = 1} , Atpρ(Q) = {(α, β, γ) ∈ Atp(Q) | 1α = 1} . For instance, say, (α, β, γ) ∈ Atpλ(Q). For all x ∈ Q, xα = xα · 1 = xα · 1β = (x1)γ = xγ. Thus α = γ. Set a = 1α. For all x ∈ Q, xα = (1x)α = 1α · xβ = a · xβ Thus α = βLa, and so every element of Atpλ(Q) has the form (βLa, β, βLa) 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: Atpλ(Q) = Atp(Q) ∩ {(βLa, β, βLa) | β ∈ Sym(Q), a ∈ Q} , Atpμ(Q) = Atp(Q) ∩ {(γR−1 c\1, γL −1 c , γ) | γ ∈ Sym(Q), c ∈ Q} , Atpρ(Q) = Atp(Q) ∩ {(α, αRb, αRb) | α ∈ Sym(Q), b ∈ Q} . 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.