A class of d-dimensional flag transitive translation planes
The curious title of this paper does not refer to planes of dimension larger than 2... In fact the authors study several cases of the classical construction of (affine) translation planes by means of spreads [see for instance \textit{D. R. Hughes} and \textit{F. C. Piper}, Projective Planes (1973; Zbl 0267.50018), Ch. VII]; each component of the spread is a d- dimensional subspace of a 2d-dimensional vector space over GF(q), where \(q=p^ r\) is a prime power. In this paper, d is an odd positive integer such that \((q-1,d)=1;\) under certain conditions imposed to the spread, it yields a plane which has a subgroup of collineations acting transitively on the set of flags (pairs of incident point line), and which is in certain cases desarguesian, in other cases non-desarguesian. The planes of order 27 and 125 previously found by the Raos are particular cases of those constructed in this paper. Reviewer's comments: (i) The reader will easily correct a few misprints (re-establishing ``non-desarguesian'' in the first line, ``Let \(\alpha\) '' in 1.12 of p. 140, the symbol for intersection in 1.3 of p. 141), but others may be hidden. (ii) The proofs are difficult to follow particularly because some notations (``M(V)'', ``D(V)'', ``E(V)'') are not defined while not being of general use in the field. (iii) It is nowhere said that the paper is about affine planes...
scientific article; zbMATH DE number 4090258
scientific article