This paper investigates the subset •(n,q) of the general linear group Gl(n+1) consisting of all elements which have no nontrivial fixed points. In particular it shows that there is a bijection between this set and the set of n-dimensional subspaces of the projective space PG(2n+1,q) which satisfy the intersection property of being skew to three given spaces which are themselves pairwise disjoint. The paper obtains this bijection by associating to each A∈•(n,q) first the row space of the row echelon matrix [IA] and then the projectivization of this space obtained by identifying scalar multiples and discarding the vector 0. Classical results for PG(3,q) then provide the basis for a recurrence relation which expresses •(n,q) in terms of •(n-1,q) and •(n-2,q). Because the development of this recurrence is constructive it provides, as an application, an efficient method for not only enumerating but for exhibiting all the nonsingular linear transformations A∈Gl(n+1) such that A+id is nonsingular as well.
