We show that the Coxeter group <mml:math alttext="$D_n$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> is the split extension of
<mml:math alttext="$n - 1$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math> copies of <mml:math alttext="$Z_2$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> by <mml:math alttext="$S_n$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> for a given action of <mml:math alttext="$S_n$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> described in the paper. We also find the centre of <mml:math alttext="$D_n$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> and some
of its other important subgroups.

The Coxeter group <mml:math alttext="$D_n$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>

The Coxeter group <svg style="vertical-align:-3.3907pt" height="15.4" version="1.1" viewBox="0 0 20.375 15.4" width="20.375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
		<g transform="matrix(.017,-0,0,-.017,.062,11.112)">
			<path id="x1D437" d="M780 372q0 -191 -178 -297q-128 -75 -335 -75h-250l7 27q61 7 77 22t28 76l77 398q11 62 0 76.5t-77 22.5l6 28h292q164 0 258.5 -67.5t94.5 -210.5zM678 375q0 119 -73 180.5t-201 61.5q-62 0 -82 -13q-17 -7 -25 -53l-80 -427q-8 -47 -4 -62t26 -20q24 -6 80 -6
q166 0 262.5 95t96.5 244z"/>
		</g>
		<g transform="matrix(.012,-0,0,-.012,13.712,15.187)">
			<path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z"/>
		</g>
	</svg>



International Journal of Mathematics and Mathematical Sciences

The Coxeter group Dn

Abstract

Copyright