Symmetric models of the real projective plane

We show that the symmetry group of a stable immersion of the real projective plane P in E3 is either trivial or is cyclic of order 3, and that of a stable map of P in E3 is conjugate to a subgroup of the full tetrahedral group. Thus Boy's surface, in its 'standard' form, is the most symmetrical stable immersion of P in E3, and Steiner's surface is given by the most symmetrical stable map of P in E3. We also construct a smooth embedding of P in E4 with symmetry group SO2 by orthogonal projection of the Veronese surface