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

Blokhuis, Aart; Jungnickel, Dieter; Schmidt, Bernhard

doi:10.1090/S0002-9939-01-06388-2

Proof of the prime power conjecture for projective planes of order $n$ with abelian collineation groups of order $n^2$
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by Aart Blokhuis, Dieter Jungnickel and Bernhard Schmidt PDF

Proc. Amer. Math. Soc. 130 (2002), 1473-1476 Request permission

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 $p$.

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