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# A note on generalized twisted fields and autotopisms of translation planes

Studia Scientiarum Mathematicarum Hungarica
Authors: Mashhour Alali and Sahem Tarawneh

The first part of this paper is a discussion of autotopism groups of certain translation planes. The second part shows that there is a certain class of semifields of order q 4 having nuclei of order q 2 , and their translation planes are not desarguesian and not isomorphic to generalized twisted field planes.

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# On B-injectors of symmetric groups S n

Studia Scientiarum Mathematicarum Hungarica
Authors: Mohammad Al-Shomrani and Mashhour. Al-Ali Bani-Ata

The aim of this paper is to describe the B-injectors of the symmetric group S n by proving the following main theorem, using a shorter proof than that followed in [1] and [3]. In this note the proof is mainly based on the minimal proof concept, and the parts we have used from these two papers are referred to.Main Theorem: Let Ω be a finite set of size n, and let BS Ω be a B-injector of S Ω. Then

1. a) If n ≢ 3 (mod 4) then B is a Sylow 2-subgroup of S Ω.
2. b) If n ≡ 3 (mod 4) then B = 〈d〉 × T where d is a 3-cycle and T is a Sylow 2-subgroup of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $C_{S_\Omega } (d)$ \end{document}.

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