The Group Action of the Dihedral Group of Order Six as Symmetries of an Equilateral Triangle

The group action of D3 as rotations and reflections of an equilateral triangle

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 15 December 2010

The group action of D3 as rotations and reflections of an equilateral triangle

D3 is the group of rotations and reflections of an equilateral triangle. We shall denote the vertices by vi, the edge connecting the vertex i to the vertex j by eij, i<j, and the face by f012. Let pij, 0i,j2, denote the point on the edge connecting vi to vj which is a third of the way from vi to vj.

v0 v2 v1 y x

Let x be the 120° counterclockwise rotation about the center taking v0 v1 v2 v0. Let y be the reflection about the line connecting vertex v0 with the midpoint of the edge e12, taking v1 v2and fixing  v0. Note that x3=1, y2=1, and yx=x-1y.

Let P= p01, p10, p12, p21, p02, p20 , V= v0, v1, v2 , E= e01, e12, e02 ,and F= f012 , denote the sets of points, vertices, edges, and faces respectively. Since D3 acts on the equilateral triangle, D3 acts on each of these sets.

Stabilizer Size of Stabilizer Orbit Size of Orbit
D3 pij= 1 1 D3Pij= P 6
D3v0= 1,y =H 2 D3v0= V 3
D3v1= 1,x2y = xHx-1 2 D3v1=V 3
D3v2= 1,xy = x2Hx-2 2 D3v2=V 3
D3e01= 1, xy = x2Hx-2 2 D3 e01=E 3
D3e12= 1, y =H 2 D3 e12=E 3
D3e02= 1, x2y = xHx-1 2 D3 e02=E 3
D3 f012 = D3 6 D3 f012 =F 1

v0 v2 v1 1 v2 v1 v0 x v1 v0 v2 x2
v0 v1 v2 y v1 v2 v0 xy v2 v0 v1 x2y


[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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