Multiplication groups of loops and projective semilinear transformations in dimension two

Abstract

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). © 2002 Elsevier Science (USA).