I am stuck on 2 problems and would appreciate any help.
1. Consider the group D3. Find the left and right cosets of H in D3 where H is the two element subgroup generated by the reflection in a line through the vertex labeled 1.
2. The rotational symmetry group G of a dodecahedron acts transitively on the set of its 20 vertices. The stabilizer of a given vertex is the three rotations (including the identity) about the line through that vertex and the centre. How many rotational symmetries does the dodecahedron have?
Apply the same argument to calculate the number of rotational symmetries
of an octahedron (the regular solid with eight faces, each an equilateral triangle).
Can anyone help with these questions? Thanks.