Question

Write out a complete Cayley table for D3.is D3 abelian?

Answer 1

For our triangle ABC I note:

1. id - identity transformation
2. Rot 120 - rotation by 120 degrees around the center of triangle
3. Rot 240 - rotation by 240 degrees around the center of triangle
4. Sym - symmetry with respect to the median passing through the point A
5. Sym 120 - symmetry with respect to the median passing through the point B
6. Sym 240 - symmetry with respect to the median passing through the point C

With these notations our Cayley table goes like this:

In particular we see that D3 is not abelian, as $sym \times rot120 \neq rot120\times sim$ .

The easiest way to study D3 is to consider the action of every element on the vertexes of ABC :

1. $id:(A,B,C)\mapsto (A,B,C)$
2. $rot120:(A,B,C)\mapsto (B,C,A)$
3. $rot240 : (A,B,C)\mapsto (C,A,B)$
4. $sym:(A,B,C)\mapsto (A,C,B)$
5. $sym120: (A,B,C)\mapsto (C,B,A)$
6. $sym240:(A,B,C)\mapsto (B,A,C)$

With this interpretation we see that $D_3 \simeq S_3$ (group of bijections of a set of 3 elements) and so it is another proof that it is not abelian (as $S_3$ is not abelian).