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

A bijective mapping phi: G --> G defined on a finite group G is complete if the mapping eta defined by eta(x) = x phi(x), x is an element of 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 = 1 mod 4 and PGL(2, q), q odd. As a consequence, we prove that in odd characteristic the projective groups PGL(n, q), n > 2, and the linear groups GL(n, q), n greater than or equal to 2, admit a complete mapping.