My question is how to find all conjugacy classes of order 5 of the rotational symmetry of the dodecahedron. I know that I could find them by considering conjugacy classes of order 5 in A5, but I would like to find them by considering the dodecahedron.
I also know that the elements of order 5 are rotations through the middle of a side of the dodecahedron. My problem is I can't visualize which rotations I have to use to prove there are two conjugacy classes of elements of order 5.
Best Answer
There are at least two conjugacy classes since the elements of order $5$ rotate through two different angles, $2\pi/5$ and $4\pi/5$.
To show that there are exactly two conjugacy classes, we need to exhibit a rotation for each pair of elements in each class that rotates the (oriented) axis of rotation of one element into that of the other.
As you say, the elements of order $5$ are rotations about the axes through the centres of the faces. We can consider each class as, say, the clockwise rotations through $2\pi/5$ and $4\pi/5$, respectively, about axes through the centre of each face, oriented towards that face.
A rotation through $\pi$ about an axis through the centre of an edge of a face swaps that face with the adjacent face with which it shares that edge. Any face can be moved to any other face by a series of swaps with adjacent faces.