- 3 6 Marks Consider The Six Element Group D6 Of All Rotational And Reflectional Symmetries Of An Equilateral Triangle 1 (64.15 KiB) Viewed 48 times
3. [6 Marks] Consider the six-element group D6 of all rotational and reflectional symmetries of an equilateral triangle in the plane: 1 3 2 Each g € D6 permutes the three vertices of this triangle, as labelled above, and this allows us to identify Do with the symmetric group S3. For example, the element of De which rotates anticlockwise by an angle of 120° corresponds to the 3-cycle (132). a) Describe (using words or pictures) the elements of D6 which correspond to (12) and (1 3) in S3. b) Consider the two orientations of the triangle: By identifying S3 with D6, and considering how elements of De act on this set of orientations, write down a homomorphism : S3 → S₂. What is the kernel of ?