The correlations of finite Desarguesian planes, Part I: Generalities

The polarities of Desarguesian planes have long been known. This author has undertaken to classifythe correlations of finite Desarguesian planes in general. In [6] we have presented all the correlations with identitycompanion automorphism which are not polarities, of these planes. In this sequence of papers, we classify thecorrelations of planes of order $ p^{2^{i}(2n+1)}, n neq 0 $, with companion automorphism ( $p^{2^{i}t}$ ), p an odd prime, $ t neq 0 $.This represents a complete classification of the correlations of planes of odd nonsquare order (i = 0). Some ofthe correlations of planes of odd square order ($ t neq 0 $ ) are also covered by the present analysis.When the companion automorphism is not trivial, the problem, naturally, becomes more involved, and a great dealbegins to hinge upon the order of the plane being odd or even, and also a square or a nonsquare.The correlations of planes of order $ 2^{2^{i}(2n+1)}, n neq 0 $, with companion automorphism $ 2^{2^{i}t}, t neq 0 $, and especiallythose of planes of order $ p^{2^{i}(2n+1)}, i neq 0 $, with companion automorphism $ p^{2^{j}(2r+1)}, j > i $ require a substantiallydifferent treatment, and will be the object of separate efforts.