Answer to Question #196571 in Abstract Algebra for Saba Umer

Question

consider the dihedral group D₆ and define its action on X={1,2,3,4,5,6} ?

Answer 1

The dihedral group  D₆ gives the group of symmetries of a regular hexagon. The group generators are given by a counter-clockwise rotation through  radians and reflection in a line joining the midpoints of two opposite edges. If  denotes rotation and  reflection, we have

D₆=<x,y:x⁶=y²=1, xy=x^-1>

If we have a set X={1,2,3,4,5,6}

The action of D₆ on X is called

• transitive if for any two x, y in X there exists a g in D₆ such that g · x = y; this is not the case
• faithful (or effective) if for any two different g, h in D₆ there exists an x in X such that g · x ≠ h · x; this is the case, because, except for the identity, symmetry groups do not contain elements that "do nothing"
• free if for any two different g, h in D₆ and all x in X we have g · x ≠ h · x; this is not the case because there are reflections