2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/209362Let $Q$ be a Buchsteiner loop. We describe the associator calculus in three variables, and show that $|Q| \ge 32$ if $Q$ is not conjugacy closed. We also show that $|Q| \ge 64$ if there exists $x \in Q$ such that $x^2$ is not in the nucleus of $Q$. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpotency class 3 and possesses an abelian inner mapping group.22 pagesGroup Theory20N05text