Semifield planes of rank 2 admitting the group S-3

Описание

Тип публикации: статья из журнала

Год издания: 2019

Идентификатор DOI: 10.21538/0134-4889-2019-25-4-118-128

Ключевые слова: semifield plane, autotopism group, symmetric group, Baer involution, homology, spread set

Аннотация: 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 translatiПоказать полностьюon 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. 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 π be a semifield plane of order p2n with a kernel containing GF(pn) for prime p, and let the linear autotopism group of π contain a subgroup H isomorphic to the symmetric group S3. 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 32n with kernel GF(3n) admits no subgroups isomorphic to S3 in the linear autotopism group. Examples of semifield planes of order 16 and 625 admitting S3 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. © 2019 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.

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Издание

Журнал: TRUDY INSTITUTA MATEMATIKI I MEKHANIKI URO RAN

Выпуск журнала: Vol. 25, Is. 4

Номера страниц: 118-128

ISSN журнала: 01344889

Место издания: YEKATERINBURG

Издатель: KRASOVSKII INST MATHEMATICS & MECHANICS URAL BRANCH RUSSIAN ACAD SCIENCES

Авторы

  • Kravtsova O.V (Siberian Fed Univ, Sci Phys Math, Krasnoyarsk 660041, Russia)
  • Moiseenkova T.V (Siberian Fed Univ, Sci Phys Math, Krasnoyarsk 660041, Russia)

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