Linear autotopism subgroups of semifield projective planes

Olga V. Kravtsova ; Daria S. Skok

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 6, pp. 705-719.

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). This hypothesis is reduced to autotopism group that consists of collineations fixing a triangle. We describe the elements of order 4 and dihedral or quaternion subgroups of order 8 in the linear autotopism group when the semifield plane is of rank 2 over its kernel. 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, homology, Baer involution, Hughes' problem.
Mots-clés : autotopism

Bibliographie

[1] D.R.Hughes, F.C.Piper, Projective planes, Springer–Verlag Publ, New-York, 1973

[2] V.D.Mazurov, E.I.Khukhro (eds.), Unsolved Problems in Group Theory, The Kourovka Notebook, 19, Sobolev Inst. Math. Publ., Novosibirsk, 2018

[3] M. Biliotti, V. Jha, N.L. Johnson, G. Menichetti, “A structural theory for two-dimensional translation planes of order $q^2$ that admit collineation groups of order $q^2$”, Geom. Dedicata, 29 (1989), 7–43

[4] O.V.Kravtsova, “Semifield planes of even order that admit the Baer involution”, The Bulletin of Irkutsk State University. Series Mathematics, 6:2 (2013), 26–37 (in Russian)

[5] O.V.Kravtsova, “Semifield planes of odd order that admit a subgroup of autotopisms isomorphic to $A_4$”, Russian Mathematics, 60:9 (2016), 7–22 | DOI

[6] N.D.Podufalov, “On spread sets and collineations of projective planes, part 1”, Contem. Math., 131, 1992, 697–705

[7] O.V.Kravtsova, “Elementary abelian 2-subgroups in an autotopism group of a semifield projective plane”, The Bulletin of Irkutsk State University. Series Mathematics, 32 (2020), 49–63 | DOI

[8] O.V.Kravtsova, “Dihedral group of order 8 in an autotopism group of a semifield projective plane of odd order”, Journal of Siberian Federal University. Mathematics $\$ Physics, 15:3 (2022), 21–27

[9] O.V.Kravtsova, D.S.Skok, “The spread set method for the construction of finite quasifields”, Trudy Inst. Mat. i Mekh. UrO RAN, 28, no. 1, 2022, 164–181 (in Russian) | DOI

[10] H.Luneburg, Translation planes, Springer-Verlag Publ, New-York, 1980

[11] O.V.Kravtsova, “On automorphisms of semifields and semifield planes”, Siberian Electronic Mathematical Reports, 13 (2016), 1300–1313 | DOI

[12] N.D.Podufalov, B.K.Durakov, O.V.Kravtsova, E.B.Durakov, “On the semifield planes of order $16^2$”, Siberian Mathematical Journal, 37:3 (1996), 616–623 (in Russian)

[13] G.E.Moorhouse, “$PSL(2,q)$ as a collineation group of projective planes of small orders”, Geom. Dedicata, 31:1 (1989), 63–88

[14] O.V.Kravtsova, “2-elements in an autotopism group of a semifield projective plane”, The Bulletin of Irkutsk State University. Series Mathematics, 39 (2022), 96–110 | DOI

[15] M.J.Ganley, “Baer involutions in semi-field planes of even order”, Geom. Dedicata, 2 (1974), 499–508

[16] N.D.Podufalov, B.K.Durakov, I.V.Busarkina, O.V.Kravtsova, E.B.Durakov, “Some results on finite projective planes”, Journal of Mathematical Sciences, 102:3 (2000), 4032–4038

[17] D.R.Hughes, M.J.Kallaher, “On the Knuth semi-fields”, Internat. J. Math. $\$ Math. Sci., 3:1 (1980), 29–45

[18] O.V.Kravtsova, “Semifield planes admitting the quaternion group $Q_8$”, Algebra and Logic, 59:1 (2020), 71–81