A Non-Associative Approach to the Finite Projective Plane Conjecture
by Douglas J. Shaw
1995

Abstract

The strong form of the Finite Projective Plane (FPP) Conjecture states that if there exists an FPP of order n, then n = pa, p prime. Since an FPP of order n is equivalent to an <n2+n+1,n+1,1> Difference Set (DS), we can investigate the FPP Conjecture by investigating these sets. A binary operation, (+) may be introduced on a subset D' of such a difference set D in a natural way so that the resulting algebraic structure, <D',(+)> is a totally symmetric quasigroup. This Dissertation begins the investigation of when <D',(+) > is isotopic to a simple p- group for some p.

It is shown that <D',(+) > is isotopic to a loop, <D',(*)> , whose isotopy-invariant diagonal is entirely compatible with the conjecture. Several results in the direction of full associativity for the structure <D',(*) > are obtained. One consequence of the nature of the diagonal of <D',(*) > is a surprising "cycle- decomposition" on the elements of D, which is uncovered and analyzed, further illuminating the deep structure of D.

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