Semifield planes of rank 2 admitting the group S-3

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, Autotopism group, Baer involution, Homology, Semifield plane, Spread set, Symmetric group

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 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.

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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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