On the admissibility of some linear and projective groups in odd characteristic

Abstract

A bijective mapping \(\emptyset :G \to G \) defined on a finite group G is complete if the mapping η defined by \(\eta (x) = x\emptyset (x) \), \(x \in G \), is bijective. In 1955 M. Hall and L. J. Paige conjectured that a finite group G has a complete mapping if and only if a Sylow 2-subgroup of G is non-cyclic or trivial. This conjecture is still open. In this paper we construct a complete mapping for the projective groups PSL\((2,q),q \equiv 1\bmod 4 \) and PGL(2,q),q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n,q GL\((n,q),n \geqslant 2 \), admit a complete mapping.