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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 nonDesarguesian 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.

Elementary Abelian 2-subgroups in an Autotopism Group of a Semifield Projective Plane} We investigate the hypotheses on a solvability of the full collineation group for non-Desarguesian semifield projective plane of a finite order (the question 11.76 in Kourovka notebook). It is well-known that this hypotheses is reduced to the solvability of an autotopism group. We study the subgroups of even order in an autotopism group using the method of a spread set over a prime subfield. It is proved that, for an elementary abelian 2-subgroups in an autotopism group, we can choose the base of a linear space such that the matrix representation of the generating elements is convenient and uniform for odd and even order; it does not depend on the space dimension. As a corollary, we show the correlation between the order of a semifield plane and the order of an elementary abelian autotopism 2-subgroup. We obtain the infinite series of the semifield planes of odd order which admit no autotopism subgroup isomorphic to the Suzuki group $Sz(2^{2n+1})$. For the even order, we obtain the condition for the nucleus of a subplane which is fixed pointwise by the involutory autotopism. If we can choose such the nucleus as a basic field, then the linear autotopism group contains no subgroup isomorphic to the alternating group $A_4$. 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 obtained results are consistent with the examples of 3-primitive semifield planes of order~81, and also with two well-known non-isomorphic semifield planes of order~16.

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A5 and find a series of semifield planes of odd order not admitting A5. [ABSTRACT FROM AUTHOR]

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A5 and find a series of semifield planes of odd order not admitting A5. [ABSTRACT FROM AUTHOR]

It is well-known that the constructions and classification of non-Desarguesian projective planes are closely connected with ones for quasifields. We consider the problems on structure of finite quasifields and semifields: automorphisms and autotopisms, maximal subfields and their orders, the spectrum of orders of non-zero elements and hypotheses about generated subsets of the multiplicative loop. [ABSTRACT FROM AUTHOR]

We develop an approach to constructing and classification of semifield projective planes with the use of a linear space and a spread set. We construct a matrix representation of the spread set of a semifield plane of odd order that admits a Baer involution in the translation complement or a subgroup of autotopisms isomorphic to the alternating group A . We give examples of semifield planes of order 81 satisfying the above indicated condition. [ABSTRACT FROM AUTHOR]

We present a description of maximal partial ovoids of size [ABSTRACT FROM AUTHOR]

Semifields 6-dimensional over the center $${\mathbb F_q}$$, having at least one nucleus of order q and at least one of the remaining nuclei of order q, are partitioned into non-empty mutually non-isotopic classes. Also, for each class a canonical form for the multiplication is presented. [ABSTRACT FROM AUTHOR]

In this paper we show that starting from a symplectic semifield spread $${\mathcal{S}}$$ of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of $${\mathcal{S}}$$, and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) ( q = s, s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125-133, ). [ABSTRACT FROM AUTHOR]

Abstract: In this paper we obtain a new description of the translation dual of a semifield introduced in [G. Lunardon, Translation ovoids, J. Geom. 76 (2003) 200–215]. Using such a description we are able to prove that a semifield and its translation dual have nuclei of the same order. Combining the Knuth cubical array and the translation dual, we give an alternate description of the chain of twelve semifields in the table of [S. Ball, G.L. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (2007) 117–129]. [Copyright &y& Elsevier]