Annals of Combinatorics 1 (1997) 227-243

On the Maximum Number of Fixed Points in Automorphisms of Prime Order of 2-(v,k,1) Designs

D.L. Kreher, D.R. Stinson, and L. Zhu

Department of Mathematical Sciences, Michigan Technological University Houghton, MI 49931, USA

Computer Science and Engineering Department, University of Nebraska Lincoln, NE 68588, USA

Department of Mathematics, Suzhou University, Suzhou 215006, P.R. China

Received January 15, 1997

AMS Subject Classification: 05B05

Abstract. In this paper, we study 2-(v,k,1) designs with automorphisms of prime order p, having the maximum possible number of fixed points. We prove an upper bound on the number of fixed points, and we study the structure of designs in which this bound is met with equality (such a design is called a p-MFP(v,k)). Several characterizations and asymptotic existence results for p-MFP(v,k) are obtained. For (p,k) = (3,3), (5,5), (2,3) and (3,4), necessary and sufficient conditions on v are obtained for the existence of a p-MFP(v,k). Further, for 3 $\leq$ k $\leq$ 5 and for any prime p$\equiv$ mod k(k-1), we establish necessary and sufficient conditions on v for the existence of a p-MFP(v,k).

Keywords: 2-design, automorphism, fixed point


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