On B-injectors of symmetric groups Sn

Abstract

The aim of this paper is to describe the B-injectors of the symmetric group Sn 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 B ≦ SΩbe a B-injector of SΩ. Then a) If n ≢ 3 (mod 4) then B is a Sylow 2-subgroup of SΩ. 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}.