A property of the groups $P\Gamma L(m,\,q)$, $q\geq 5$

It is well known that the existence of a sharply doubly transitive set of permutations is equivalent to the existence of a projective plane. The natural representation of the group $P\Gamma L(m,q),m \geqq 2$, is doubly transitive and it is proved here that this permutation group does not contain a sharply doubly transitive subset when $q \geqq 5$.