Semifield planes of rank 2 admitting the group $S_3$

O. V. Kravtsova ; T. V. Moiseenkova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 118-128

Résumé

One of the classical problems in projective geometry is to construct an object from known constraints on its automorphisms. We consider finite projective planes coordinatized by a semifield, i.e., by an algebraic system satisfying all axioms of a skew-field except for the associativity of multiplication. Such a plane is a translation plane admitting a transitive elation group with an affine axis. Let $\pi$ be a semifield plane of order $p^{2n}$ with a kernel containing $GF(p^n)$ for prime $p$, and let the linear autotopism group of $\pi$ contain a subgroup $H$ isomorphic to the symmetric group $S_3$. For the construction and analysis of such planes, we use a linear space and a spread set, which is a special family of linear mappings. We find a matrix representation for the subgroup $H$ and for the spread set of a semifield plane if $p=2$ and if $p>2$. We also study the existence of central collineations in $H$. It is proved that a semifield plane of order $3^{2n}$ with kernel $GF(3^n)$ admits no subgroups isomorphic to $S_3$ in the linear autotopism group. Examples of semifield planes of order 16 and 625 admitting $S_3$ are found. The obtained results can be generalized for semifield planes of rank greater than 2 and can be applied, in particular, for studying the known hypothesis that the full collineation group of any finite non-Desarguesian semifield plane is solvable.

Détail

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

Bibliographie

[1] Mazurov V.D., Khukhro E.I., Nereshennye voprosy teorii grupp. Kourovskaya tetrad, izd. 16-e, dop., vklyuchayuschee arkhiv reshennykh zadach, In-t matematiki im. S.L. Soboleva SO RAN, Novosibirsk, 2006, 193 pp.

[2] Podufalov N.D., Durakov B.K., Kravtsova O.V., Durakov E.B., “O polupolevykh ploskostyakh poryadka $16^2$”, Sib. mat. zhurn., 37:3 (1996), 616–623 | MR | Zbl

[3] Jha V., Johnson N.L., “The translation planes of order 81 admitting SL(2,5)”, Note di Matematica, 24:2 (2005), 59–73 | DOI | MR | Zbl

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

[5] Levchuk V.M., Kravtsova O.V., “Problems on structure of finite quasifelds and projective translation planes”, Lobachevskii J. Math., 38:4 (2017), 688–698 | DOI | MR | Zbl

[6] Rua I.F., Combarro E.F., Ranilla J., “Classification of semifields of order 64”, J. Algebra, 322:11 (2009), 4011–4029 | DOI | MR | Zbl

[7] Kravtsova O.V., Moiseenkova T.V., “Polupolevye proektivnye ploskosti, dopuskayuschie podgruppu kollineatsii, izomorfnuyu $S_3$”, Algebra, teoriya chisel i diskretnaya geometriya: sovremennye problemy, prilozheniya i problemy istorii, materialy Mezhdunar. konf. posvyaschennoi 80-letiyu so dnya rozhdeniya professora Mishelya Deza, Izd-vo Tulskogo gos. ped. un-ta im. L.N. Tolstogo, Tula, 2019, 260–262

[8] Hughes D.R., Piper F.C., Projective planes, Springer-Verlag, N Y, 1973, 292 pp. | MR | Zbl

[9] Johnson N.L., Jha V., and Biliotti M., Handbook of finite translation planes, Chapman and Hall, Boca Raton; London; N Y, 2007, 861 pp. | MR | Zbl

[10] Kravtsova O.V., “Polupolevye ploskosti, dopuskayuschie berovskuyu involyutsiyu”, Izv. Irkut. gos. un-ta. Ser. Matematika, 2013, no. 2, 26–38

[11] Vaughan, T.P., “Polynomials and linear transformations over finite fields”, J. Reine Angew. Math., 267 (1974), 179–206 | DOI | MR | Zbl

[12] Huang H. , Johnson N.L., “8 semifield planes of order $8^2$”, Discrete Math., 80:1 (1990), 69–79 | DOI | MR | Zbl