Abelian projective planes of square order (Q1118812)

The author proves that, under certain conditions, multipliers of an abelian projective plane of square order have odd order modulo \(v^*\), where \(v^*\) is the exponent of the underlying Singer group. As a consequence, he is able to establish the non-existence of an infinitive number of abelian projective planes of square order. Namely: suppose that \(m\equiv 10\) or 12 (mod 14), then, (I) there does not exist a cyclic projective plane of order \(m^{2r}\) for each r with \((r,3)=1\), and (II) there does not exist an abelian projective plane of order \(m^{2r}\) for each positive integer r. The main theorem is: Let D be a cyclic \((n^ 2+n+1,1)\) difference set in a group G. Suppose that \(n=m^{2S}\) with \((S,3)=1\). Let \(t^{n+1}\) be a multiplier of D. Suppose that there exists a prime divisor q of \(m^ 2- m+1\) such that t has odd order modulo q. Then t has odd order modulo v, where \(v=m^ 4+m^ 2+1,\) and hence t has odd order modulo p for each prime divisor p of v. Finally, the author conjectures that, if D is a projective plane of order n admitting a cyclic regular automorphism group, then n must be a prime power and D must be isomorphic to PG(2,n).