The Group Action of the Dihedral Group of Order Eight as Symmetries of a Square

The group action of D4 as rotations and reflections of a square

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

Last updates: 17 December 2010

The group action of D4 as rotations and reflections of a square

D4 is the group of rotations and reflections of a square. 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 f0123. 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 v3 v2 v1 y x

Let x be the 90° counterclockwise rotation about the center taking v0 v1 v2 v3 v0. Let y be the reflection about the line connecting vertex v0 with vertex v2, taking v1 v3and fixing  v0 and v2. Note that x4=1, y2=1, and yx=x-1y.

Let P= p01, p10, p12, p21, p23, p32, p03, p30 , V= v0, v1, v2, v3 , E= e01, e12, e23, e03 ,and F= f0123 , denote the sets of points, vertices, edges, and faces respectively. Since D4 acts on the square, D4 acts on each of these sets.

Stabilizer Size of Stabilizer Orbit Size of Orbit
D4 pij= 1 1 D4pij= P 8
D4v0= 1,y =H 2 D4v0= V 4
D4v1= 1,x2y = xHx-1 2 D4v1=V 4
D4v2= 1,y = H 2 D4v2=V 4
D4v3= 1,x2y = xHx-1 2 D4v3=V 4
D4e01= 1, xy = J 2 D4 e01=E 4
D4e23= 1, xy =J 2 D4 e23=E 4
D4e12= 1, x3y = xJx-1 2 D4 e12=E 4
D4e03= 1, x3y = xJx-1 2 D4 e03=E 4
D4 f012 = D4 8 D4 f0123 =F 1

1 v0 v3 v2 v1 x v3 v2 v1 v0 x2 v2 v1 v0 v3 x3 v1 v0 v3 v2
y v0 v1 v2 v3 xy v1 v2 v3 v0 x2y v2 v3 v0 v1 x3y v3 v0 v1 v2


[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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