The classification of the translation planes of order 16, I

Abstract

The translation planes of order 16 are completely classified. The exceptional isomorphism A s -~ GLI4. 2)gives a crucial computational approach to this problem * The authors thank the Deutsche Forschungsgemeinschaft for their support of this research.

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“…The planes of type DP c are translation planes. For their classification we refer the reader to [DR83]. 4.2.…”

“…For type DP b , the group of translations is calculated to determine if a plane is a translation plane. By [Roy], the translation planes of order 16 are uniquely determined by the number of Fano-subplanes (all translation planes of order 16 are known by [DR83]). For the remaining case, explicit isomorphisms are constructed.…”

“…The translation planes of order 16 have been classified in [DR83], i.e. only types different from DP c have to be considered.…”

Elation Laguerre planes of order 16 are ovoidal

“…The translation planes of order 16 were classified by Dempwolff and Reifart [5]. There are eight distinct translation planes of order 16:…”

Large doubly transitive orbits on a line

“…Thus SL(2, 7) < G and n = 16. A direct inspection of the list of the full collineation groups of all translation planes of order 16 given in [16] (see also [53]) rules out this case. So, the cases (2b) and (3b) of Theorem 1.1 cannot occur when n is the projective extension of a translation plane of orderrc.…”