Frobenius Collineations in Finite Projective Planes

of order n on V . It follows that R induces a projective collineation φ on the (n−1)dimensional projective space PG(n−1, q). We call φ and any projective collineation conjugate to φ a Frobenius collineation. In the present paper we shall study the case n = 3, that is, the Frobenius collineations of the projective plane PG(2, q). Let P = PG(2, q). Then every Singer cycle σ (see Section 3) of P defines a partition P(σ) of the point set of P into pairwise disjoint Baer subplanes. These partitions are called linear Baer partitions or, equivalently, Singer Baer partitions [17]. If % is a Frobenius collineation of P , then we define E% to be the set of Baer subplanes of P fixed by %. It turns out that for q ≡ 2 mod 3 we have |P(σ)∩E%| ∈ {0, 1, 3} with |P(σ) ∩ E%| = 3 if and only if % ∈ NG(< σ >), where G = PGL3(q) (see 3.5).