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Automorphisms of 2-(22, 8, 4) designs

✍ Scribed by I. Landgev; V.D. Tonchev

Publisher
Elsevier Science
Year
1989
Tongue
English
Weight
793 KB
Volume
77
Category
Article
ISSN
0012-365X

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✦ Synopsis

Dedicated to Professor Hahn Hanani on the occasion of his 75th birthday.

It is shown that a 2- (22,8,4) design cannot possess any nontriviai automorphisms of an odd order. k = 8, A = 4. This is the smallest case left open in Table 5.23 of the remarkable Hanani's article [7]. Many of the open problems from that table have been resolved during the last decade, some of then by Professor Hanani himself (cf. Mathon and Rosa ill]). However, the existence of the smallest and most challenging 2-(22,8,4) design is still in doubt.

In this paper we investigate possible automorphism groups of a design with such parameters and show that if one exists, its full automorphism group must be either a 2-group, or trivial. Our method is based on examination of possible orbit structures of cyclic automorphism groups of a prime order by use of tactical decompositions.

An essential case of automorphisms of order 3 fixing exactly one point has been recently investigated by Kapralov [9], who found all (exactiy 53) possible orbit structures and showed (partially by computer) that none of those yields a design. We show in this paper that for an odd prime order automorphism of any other type, there is no possible orbit structure at all. Our proof does not involve any computer computations.

2. Preliminaries

We asssume that the reader is familiar with the basic notions and facts from design theory (cf. e.g. [3, 4, 5, 8, 131).

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