How would one prove the dihedral group D_n is a group?

[ry22]

I don't understand how to show that the reflections and rotations are associative. Thanks for any help.

 

[SteveL27]

I don't understand how to show that the reflections and rotations are associative. Thanks for any help.

Composition of bijections from a set to itself is associative. That's a handy principle to know, because it shortcuts the need to prove special cases. Instead of trying to visualize rotations and reflections, all you have to do is note that each reflection or rotation is permutation of the vertices.

Say A is a set, and consider the set S(A) of all bijections of A to itself. You can also think of these as permutations of the elements of the set.

If you compose two bijections you get another bijection (must be proved). And if f, g, and h are bijections, then (fg)g = f(gh) where "fg" means "f composed with g," often denoted f o g. So we could also say that (f o g) o h = f o (g o h).

You should prove that. Once you do, then any time you have a collection of geometric transformations, you know that their compositions are associative.

 

[ry22]

Thanks man! I got it!