How do you show that S3 is isomorphic to the symmetries on a

How do you show that S₃ is isomorphic to the symmetries on an equalateral triangle?

Solution

if we look at the elemets of D3 we see that there rotations act on the vertexes of the triangle as 3-cycles act on the elements of {1, 2, 3}, while reflections exchange two vertexes and leave the other fixed, therefore they act as 2-cycles on the elements of {1, 2, 3}.
Then if you denote with (A, B, C) the rotation of D3 sending the vertex A to B, B to C and C to A (that is the rotation of 120° around the center of the triangle) and with (A, B) the symmetry around the axis of the side AB you see that the map sending
(A, B, C) |-----> (1, 2, 3)
(A, B, C)*(A, B, C) = (A, C, B) |------> (1, 2, 3)*(1, 2, 3) = (1, 3, 2)
(this one is the rotation of 240°)
(A, B) |-----> (1, 2)
(B, C) |-----> (2, 3)
(A, C) |-----> (1, 3)
id |----> id
is an isomorphism.