chapter  8
10 Pages

Permutation Groups

WithVictor E. Hill, Thomas T. Read

In Examples 1.2, 1.3, and 4.5 we introduced groups D 3, Z 6, and D 4 which entailed symmetries of the trigonal bipyramid, the hexagon, and the square. More generally, Example 4.4c introduced the group Dn of the regular polygon having n sides. (Note that the group Z 6 of rotations of the hexagon is a proper subgroup of the full group D 6 of symmetries of the regular hexagon.) In the notation of Chapter 1 we viewed these groups as permutations among the vertices of the figure. Comparing Exercises 1.4 and 3.4 with Example 4.5, we have r = (1234) and c = (14)(23), which generate the whole of D 4; here we have expressed the group terms of its action on the set of vertices, although, of course, we could have considered the sides of the square instead. (See Exercises 8.1, 3.12, and 3.13.)