Question

Let G be the group of symmetries of an equilateral triangle. Recall that G has 6 elements: the identity symmetry e, two nontrivial rotations, and three flips. Let r be be a rotation 120^o counter-clockwise, so that the two nontrivial rotations in G are r and r². To keep track of the flips, let A be the lower left vertex of the triangle, B the top vertex, and C the lower right vertex of the triangle; and write f_a, f_b, or f_C for the flip that leaves vertex A, B, C stable respectively.

(a) Give the group table for G.

(b) Draw a subgroup diagram for G.

(c) Is G isomorphic to Z₆? If yes, give a function Z₆ → G defining the isomorphism. If not, explain why not.

Answer 1