Whitesides, S. H. Fixed point free collineations of order 7 in projective planes of order 9. (English) Zbl 0599.51009 Algebras Groups Geom. 2, 564-578 (1985). The main result is that the only projective plane of order 9 with a fixed point free collineation of order 7 is the desarguesian plane. Taken together with other results from the literature, this means that the full automorphism group of any undiscovered projective plane of order 9 must have order a power of 3. The paper develops in a constructive way a matrix technique that has generally been used to obtain non-existence results. The constructive methods presented here are powerful enough to construct projective planes and can be used for planes of any order. The proof of the main result involves a computation that can be done (with some difficulty) by hand but that can be broken into simple steps for fast machine solution. The computation reconstructs the desarguesian plane. MSC: 51E15 Finite affine and projective planes (geometric aspects) Keywords:reduced incidence matrix; projective plane of order 9; fixed point free collineation; desarguesian plane PDFBibTeX XMLCite \textit{S. H. Whitesides}, Algebras Groups Geom. 2, 564--578 (1985; Zbl 0599.51009)