Homework for Week 7

Part 1

1. (Rotman 2.67.iv) Prove that two groups that are each isomorphic to a third group are isomorphic to each other.
2. (Rotman 2.68) Prove that a group is abelian if and only if the function f from G to itself given by f(a) = a^-1 is a homomorphism.
3. Prove that a group homomorphism f from G to H is injective exactly when Ker(f) is just the identity.
4. Prove that a group homomorphism f from G to H is surjective exactly when Im(f) is all of H.
5. Prove that if |G| and |H| are distinct primes and f is a group homomorphism from G to H then f is the trivial homomorphism (meaning f(g)=e for all g in G).
6. The image of a cyclic group is cyclic: prove that if f is a group homomorphism from G to H and G is a cyclic group then the image of f is a cyclic subgroup of H.
7. The image of an abelian group is abelian: prove that if f is a group homomorphism from G to H and G is abelian then Im(f) is an abelian subgroup of H.

Part 2

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 1: Due Tuesday 3/8
• Part 2: Due Friday 3/11