Prove that the set of symmetries of a figure F in the plane forms a group
how do i define each element of the set ?
Here's one way...
Let be the set of vertices of . Then you can think of the group of symmetries as bijections (permutations) of . Your law of composition is then function composition.
For example, if your figure is a square, then let , where the positions of 1,2,3,4 in the plane are (1,1), (-1,1), (-1,-1), (1,-1), respectively. What are the symmetries of the square? We have rotations and reflections:
Rotations (counter clockwise):
,
,
.
Reflections:
through x-axis ,
through y-axis ,
through (2,4)-diagonal ,
through (1,3)-diagonal ,
and of course we can leave the poor square alone=identity.
The set of vertices can also be changed to the set of edges of the figure, and when in >2 dimensions, you can consider the set of faces too. There are other ways to define such a group, but I find this the most intuitive.