Doubly transitive sets of permutations characterising projective planes

✍ Scribed by R. P. Burn

• Publisher: Springer
• Year: 1973
• Tongue: English
• Weight: 295 KB
• Volume: 2
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00149282

✦ Synopsis

If U and V are distinct points of a projective plane and I is a line not through U or V, then to each/, there corresponds a unique mapping, 2, of the pencil of lines through U onto the pencil through V such that for any line u, U su, u c~ u2 eL if a set R is used to label the lines through U, and the same set is used to label the lines through V, omitting in each case the line UV, then 2 induces a permutation on R.

Witt [8] was the first to use the permutations, 2, to investigate projective planes, using them to establish the uniqueness of the planes of small order.

Hall [2] proved that the set S of all permutations 2 satisfies (i) S is doubly transitive on R, (ii) if ~, fl e S, ~ # fl, then c~fl-1 stabilises at most one element of R, (iii) if x, y ~R and ~sS such that x~ #y, there exists a unique fl with xfl=y and ~fl-1 stabilises nothing.

Conversely, Hall showed that these conditions were sufficient to ensure that the set S was the set of perspectivities of the type we have described for some plane. Under the further assumption that 1 ~S, Hall showed that S is a group if and only ifR is a near-field under the ternary operation.

Havel [3] has characterised those S which make R a quasi-field with one inverse property.