2-elements in an autotopism group of a semifield projective plane

Olga V. Kravtsova

The Bulletin of Irkutsk State University. Series Mathematics, Tome 39 (2022), pp. 96-110

Résumé

We investigate the well-known hypothesis of D.R. Hughes that the full collineation group of non-Desarguesian semifield projective plane of a finite order is solvable (the question 11.76 in Kourovka notebook was written down by N.D. Podufalov). The spread set method is used to construct the semifield projective planes with cyclic 2-subgroup of autotopisms in the case of linear space of any dimension over the field of prime order. This study completes the analogous considerations of elementary abelian 2-subgroups. We obtain the natural restriction to the order of 2-element for the semifield planes for odd and even order. It is proved that some projective linear groups can not be the autotopism subgroups for the infinite series of semifield planes. The matrix representation of Baer involution allows us to define the geometric property of autotopism of order 4. We can choose the base of a linear space such that the matrix representation of these autotopisms is convenient and uniform, it does not depend on the space dimension. The minimal counter-example is constructed to explain the restriction to the plane order. As a corollary, we proved the solvability of the full collineation group when the non-Desarguesian semifield plane has a certain even order and all its Baer subplanes are also non-Desarguesain. The main results can be used as technical for the further studies of the subgroups of even order in an autotopism group for a finite non-Desarguesian semifield plane. The results obtained are useful to investigate the semifield planes with the autotopism subgroups from J.G. Thompson's list of minimal simple groups.

Détail

Keywords: semifield plane, spread set, Baer involution, homology
Mots-clés : autotopism.

Bibliographie

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