Write out a complete Cayley table for D3.is D3 abelian?
For our triangle ABC I note:
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 :
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).
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