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Let N be a nilpotent group normal in a group G. Suppose that G acts transitively upon the points of a nite non-Desarguesian projective plane P. We prove that, if P has square order, then N must act semi-regularly on P. In addition we prove that if a nite non-Desarguesian projective plane P admits more than one nilpotent group which is regular on the points of P then P has non-square order and the automorphism group of P has odd order. | Nilpotent Singer groups Nick Gill 9 Leonard Road Redfield Bristol BS5 9NS UK. nickgill@cantab.net Submitted Jun 11 2006 Accepted Oct 10 2006 Published Oct 27 2006 Mathematics Subject Classification 20B25 51A35 Abstract Let N be a nilpotent group normal in a group G. Suppose that G acts transitively upon the points of a finite non-Desarguesian projective plane P. We prove that if P has square order then N must act semi-regularly on P. In addition we prove that if a finite non-Desarguesian projective plane P admits more than one nilpotent group which is regular on the points of P then P has non-square order and the automorphism group of P has odd order. 1 Introduction A Singer group S of a projective plane P of order x is a collineation group of P which acts sharply transitively on the points of P. The existence of such a Singer group is equivalent to a v k 1 difference set in S where v x2 x 1 k x 1 and the associated 2 v k 1 design is isomorphic to P. Ho Ho98 theorem 1 has proved the following theorem concerning abelian Singer groups Theorem C. A finite projective plane which admits more than one abelian Singer group is Desarguesian. We will present an alternative proof of this theorem our proof unlike Ho s will be dependent on the Classification of Finite Simple Groups and then will present work aimed at extending the result to nilpotent Singer groups. In particular we prove the following Theorem B. Suppose that a non-Desarguesian finite projective plane P of order x admits more than one nilpotent Singer group. Then the automorphism group of P has odd order and x is not a square. I wish to thank the University of Western Australia and the University of Gent for their support during the writing of this paper. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R94 1 In the course of proving Theorem B we will need to prove the following Theorem A. Let F be a nilpotent group which is normal in a transitive automorphism group G of P a projective plane of order x u2. Then F
