Proof of the prime power conjecture for projective planes of order 𝑛 with abelian collineation groups of order 𝑛²

Abstract

Abstract. Let G be an abelian collineation group of order n 2 of a projective plane of order n. We show that n must be a prime power, and that the p-rank of G is at least b + 1 if n = p b for an odd prime p.

Polarities of shift planes

“…The study of shift planes via relative difference sets seems to be particularly useful in the finite case, see e.g. [18], [32], [8], or [4]. 2 The normalizer of the shift group 2.1 Definition.…”

“…There remains the possibility L(0) = {0, a, b, −a − b}. Since there exists a shift plane with |∆| = 24 (namely, the plane over the field with 4 elements, see 1.8) and since the shift group in that case is (Z/4Z) 2 by 5.8, the last remaining choice indeed yields a shift plane.…”

‐Relative Difference Sets and Their Representations

“…Theorem Let D be a ‐RDS in an abelian group G of order q 2 . If q is even, then q is a power of 2 (say ), and the forbidden subgroup (see also for a short proof); If q is odd, then q is a prime power (say ) and the rank of G , i.e. the smallest cardinality of a generating set for G , is at least . …”

“…Let Z m denote the cyclic group of order m. Ganley [14] and Blokhuis, Jungnickel, Schmidt [3] proved the following result: Theorem 1.1. Let D be a (q, q, q, 1)-RDS in an abelian group G of order q 2 .…”

“…• [14] If q is even, then q is a power of 2 (say q = 2 n ), G ∼ = Z n 4 and the forbidden subgroup N ∼ = Z n 2 (see also [21] for a short proof ); • [3] If q is odd, then q is a prime power (say q = p n ) and the rank of G, i.e.…”