Multiplication Groups of Loops and Projective Semilinear Transformations in Dimension Two

For a loop Q denote by Mlt Q the permutation group generated by translations x xa and x ax, a Q. If Q is defined on F {∞}, where F is a finite field with at least five elements, and if Mlt Q ≤ PΓL(2, F), then Q is an abelian group. The same result holds for an infinite field F, if the automorphisms associated with translations generate a finite subgroup of Aut(F).