Homework for Week 7

Part 1

  1. Your RSA public key is n=35 and e=11.
    1. Send yourself the message 21. What is the encrypted message?
    2. Find the additional information you need to decode your message.
    3. You receive the encrypted message 13. What message was sent?
  2. Anatoly's RSA public key is n=35 and e=11.
    1. Send Anatoly the message 12. What is the encrypted message?
    2. Help Anatoly by finding the additional information to decode your message.
    3. You receive the encrypted message 28. What message was sent?
  3. Every group of order p is cyclic: If G is a group of order p (where p is prime), then prove that G is cyclic.
  4. If G is a group of order p2 (where p is prime) then prove that G has an element of order p.
  5. (Rotman 2.64, i-vi) True or false with reasons:
    1. If G and H are additive groups, then every homomorphism f from G to H satisfies f(x+y) = f(x)+f(y) for all x,y in G.
    2. A function f from R to Rx is a homomorphism if and only if f(x+y) = f(x)+f(y) for all x,y in R. (Here R denotes the real numbers and Rx denotes the nonzero real numbers.)
    3. The inclusion that maps Z into R is a homomorphism of additive groups.
    4. The subgroup {0} of Z is isomorphic to the subgroup {(1)(2)(3)(4)(5)} of S5.
    5. Any two finite groups of the same order are isomorphic.
    6. If p is a prime, any two groups of order p are isomorphic.
  6. Prove that homomorphisms send inverses to inverses, in other words that f(g-1) = f(g)-1. (In words, the inverse of g in G is sent to the inverse of f(g) in H.)
  7. (Rotman 2.67.i) Prove that the composite of homomorphisms is itself a homomorphism.

Part 2

  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.