# homomorphism between diedral group $D_3$ triangle isometries and $S_3$ identification problem

My question deals with the dihedral group $D_3$ of equilateral triangle 123 (1 top vertex, 2 bottom right vertex, 3 bottom left vertex).

• R1 is the counterclockwise rotation of 120 degrees.

• R2 is the counterclockwise rotation of 240 degrees.

• SA is the symmetry through the top angle bissectrice (exchanging vertex 2 and 3)

• SB is the symmetry through the right angle bissectrice (exchanging vertex 1 and 3)

• SC is the symmetry through the left angle bisssectrice (exchanging vertex 1 and 2)

I constructed the Cayley table accordingly, and compared it to the $S_3$ Cayley table: they match very well, hence my deduction that:

• R1 matches with permutation (123) of $S_3$.

• R2 matches with permutation (321).

• SA matches with permutation (23).

• SB matches with permutation (12).

• SC matches with permutation (31).

First question, is this correspondence fully right? The shapes of the two tables are quite similar, but I did not try all the possibilities.

Second problem :

I tried to deepen the above correspondence as follows :

If I write a symmetry in a line with 1st figure = top vertex, 2nd figure = right vertex and 3rd figure = left vertex, the symmetry R1 can be written

The group $S_3$ acts on the set with three points. If you number the vertices of an equilateral triangle 1,2 and 3 then you can see that $S_3$ acts on your triangle. Looking at the action of $S_3$ on your triangle, you can see that you get all of the symmetries of the triangle. As the group of symmetries of the triangle is precisely $D_3$, you are done!$^{\dagger}$
$^{\dagger}$Well, actually this just shows that there is a surjective homomorphism $S_3\rightarrow D_3$. Do you understand why this implies that there is an isomorphism between these groups?