How do I check this simple set is an Abelian group? The n-gon in question is a 3-gon. It is an equilateral triangle to be exact.
This is a Dihedral group of order 6 (3 reflections and 3 rotations)
I have plotted the Cayley's table.
The set of elements in $$D_3$$ is 
$${R_0,R_{120},R_{240},F,F',F"}$$
I know that in order to show this set is an Abelian group, 4 properties must be satisfied:
1) Closure (but under what binary condition?)
2)Identity
3)Inverse
4)Commutativity
How should I apply to property test? 
 A: (Just so there's an answer)
A general fact that you can use to test whether a binary operation $\ast$ on a set $S$ is commutative given the Cayley table has already been computed is by seeing whether or not it is symmetric about the main diagonal (i.e. the one going from the upper-left corner to the lower-right corner).  This fact follows immediately from the definition of what it means to be commutative and from what the Cayley table is.
As far as the identity element is concerned, remember that for a function $f:S\rightarrow S$, we have $\mathrm{id}_S\circ f=f\circ \mathrm{id}_S$, so that once you've found that the identity map is in your set, then you have an identity for functional composition.
For inverses, the best idea is to try to think of $D_3$ geometrically as the set of rotational symmetries for the equilateral triangle.  Your elements consist of rotations about the center and composition with a reflection about one of the (fixed) axes of symmetry.  What would you think of as the inverse of a rotation about some angle $\theta$?  Would would you think of as the inverse of a reflection?  Note that once you've found $x^{-1}$ and $y^{-1}$ for some elements $x,y$, the inverse of $xy$ is given by $y^{-1}x^{-1}$, allowing you to compute the inverses of the compositions once you've calculated the inverses of the reflection and of the rotations.  
Finally, for associativity, all you need to realize that functional composition is always associative.
