Homework for Week 8

Part 1

  1. Is the subgroup H normal in G? (Explain briefly.)
    1. H = 〈 2 〉 , G = Z/10Z
    2. H = 〈 (12)(3) 〉 , G = S3
    3. H = 〈 (123) 〉, G = S3
  2. For each normal subgroup in the previous problem, write the "multiplication" table of G/H (with respect to the group operation).
  3. Prove that if H is a normal subgroup of G then the map f given by f(g) = g * H is a group homomorphism from G to the quotient group G/H. What is the kernel of the homomorphism?
  4. Prove that if G is an abelian group, then any quotient group of G is abelian.
  5. Prove that if G is a cyclic group, then any quotient group of G is cyclic.
  6. Prove or give a counterexample: If K is a normal subgroup of H, and if H is a normal subgroup of G, then is K a normal subgroup of G?

Part 2

  1. (Rotman 1.37) True or false, with reasons.
    1. If a finite group G acts on a set X then X must be finite.
    2. If a group acts on a finite set X then G must be finite.
    3. If a group G acts on a set X and if x,y are in X then Stab(x) is isomorphic to Stab(y).
    4. If a group G acts on a set X, and if x,y are both in the same orbit in X, then Stab(x) is isomorphic to Stab(y).
  2. (Rotman 1.24, parts) Try to relate this to group actions!
    1. How many permutations in S5 commute with (12)(34)?
    2. How many permutations in S7 commute with (12)(345)?
    3. List all of the permutations in S7 that commute with (12)(345)?
  3. How many rotational and reflectional symmetries of the pentagon are there?
  4. How many rotational symmetries does a cube have?
  5. (Recommended) The first problem from week 9: how many rotational symmetries does a dodecahedron have?