On a theorem of Rigby

Abstract

Let ${\Delta = BAG(2, q)}$ denote the classical biaffine plane of order q, that is, the symmetric ${((q^2 - 1)_q)}$ configuration obtained from the classical affine plane ${\Sigma = AG(2, q)}$ of order q by omitting a point of ${\Sigma}$ together with all lines through this point. Now let ${q \geq 4}$ be a power of a prime p and assume that ${\Delta}$ admits an embedding into the projective plane ${\Pi = PG(2, F)}$, where F is a (not necessarily commutative) field. Then this embedding extends to a projective subplane ${\Pi_0 \cong PG(2, q)}$ of ${\Pi}$; in particular, F has characteristic p. Consequently, ${BAG(2, q)}$ with ${q\geq 4}$ admits an embedding into ${PG(2, q')}$ if only if q′ is a power of q. This strengthens a result of Rigby (Canad J Math 17:977–1009, 1965) in a special case while simultaneously providing a more elegant proof.