Plane Isometries

MikeDietrich

1. The problem statement, all variables and given/known data
Draw two plane figures, each having a 12 element group of symmetries, such that the two groups are NOT isomorphic. Demonstrate that they are not isomorphic.


2. Relevant equations
I know that every finite group of isometries of the plane is isomorphic to either Z_n or to the dihedral group D_n.



3. The attempt at a solution
I drew a regular hexagon (D_6) but now I am stuck as to what to draw for a figure to represent Z_12. Would a 12 bladed windmill (pinwheel) type shape with pronged ends work?

ystael

Yes, this will work, since it is not preserved by a reflection.