Department of Mathematics, Kuwait University, P. O. Box 5969, 13060 Safat, Kuwait, e-mail: farran@math-1.sci.kuniv.edu.kw;Centro de Matematica da Universidade do Porto, Faculdade de Ciencias, 4050 Porto, Portugal, e-mail: mspinto@fc.up.pt;

Faculty of Mathematical Studies, University of Southampton, Southampton SO17 3BJ, U. K., e-mail: sar@maths.soton.ac.uk

**Abstract:** We show that the symmetry group of a stable immersion of the real projective plane $P$ in $E^3$ is either trivial or is cyclic of order $3$, and that of a stable map of $P$ in $E^3$ 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 $E^3$, and Steiner's surface is given by the most symmetrical stable map of $P$ in $E^3$. We also construct a smooth embedding of $P$ in $E^4$ with symmetry group $SO(2)$ by orthogonal projection of the Veronese surface.

**Classification (MSC91):** 57R42, 57R40

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