Elementary Abelian 2-subgroups in an Autotopism Group of a Semifield Projective Plane

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.