On the finite projective planes of order up to q4, q odd, admitting PSL(3,q) as a collineation group

In this paper, it is shown that any projective plane Π of order n ≤ q 4 , q odd, that admits a group G ∼ = PSL(3, q) as a collineation group contains a G-invariant Desarguesian subplane of order q.Moreover, the involutions and suitable p-elements in G are homologies and elations of Π, respectively.In particular, if n ≤ q 3 , actually, n = q, q 2 or q 3 .