On abelian projective planes (Q1062011)

In this paper the author uses character theory and the notion of discriminant to prove the following nonexistence condition for projective planes of non square order n, with an abelian Singer group G: ''If h is the order of a subgroup of G, and p is a prime divisor of n, then \(h(n^ 2+n+1)(-1)^{[(n^ 2+n+1)-h]/2h}\) must be a square in the ring of p- adic numbers''. This condition is stronger than all known results in this direction. As an application the author considers the special case \(n\equiv 1,2(mod 4).\) For \(h=1\), the above condition yields that if n is not a square, then -1 must be a square module each odd prime divisor of n, and not only for odd prime divisors of the square free part of n (as one would get from the Bruck-Ryser-Chowla theorem). In general, the author deduces even more, namely: ''If \(n\equiv 1,2(mod 4),\) then the order h of each subgroup of an abelian Singer group must be a square modulo every odd prime divisor of n, or n must be a square''.