Answer to Question #144414 in Abstract Algebra for Ms

Question #144414

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


1
Expert's answer
2020-11-17T06:39:27-0500


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).


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