(Exercises are taken from Algebra: Abstract and Concrete, Edition 2.6 by Frederick M. Goodman.) 1. Given a polyhedron with set V of vertices of size m, consider the evident action by its symmetry group G on the set V, which gives a homomorphism o : G - Sym(V) ~ Sm. For each of the four cases (cube, octahedron, dodecahedron, icosahedron), describe the cycle type of o(9) for each type of element g E G (according to the classification of elements in G that I gave in class). 2. Let G act on a set X. Show that for x e X and ge G we have g Stab(x)g-1 = Stab(gx).3. Identify D4y as the group of rotational symmetries of the square in the zy-plane with vertices {+e1,+ea}. Thus Dy is a subgroup of SO(3). Determine the orbits of the evident action by D4 on R3. 4. Show that for a group G, the function 7: G - Sym(G) defined by 7()(z) := zg is a group action if and only if G is abelian. 5. Let n = 2k +1 be an odd integer with n > 3. Describe all the conjugacy classes in D,, (there are k+2) and determine their sizes. Pick a representative from each class. For each of these representatives, describe the elements of its centralizer group. 6. Let n =2k be an even integer with n > 4. Describe all the conjugacy classes in D,, (there are k + 3 ) and determine their sizes. Pick a representative from each class. For each of these representatives, describe the elements of its centralizer group. 7. List the conjugacy classes in S5 (there are 7 ) and determine their sizes. Pick a representative from each class. For each of these representatives, describe the elements of its centralizer group. 8. Let G be a group with normal subgroup N. Show that if o e N, then Clc(o) c N. Give an example to show that it is possible that CIN (o) + Clc(o)