Homework for Week 7
Part 1
- Your RSA public key is n=35 and e=11.
- Send yourself the message 21. What is the encrypted message?
- Find the additional information you need to decode your message.
- You receive the encrypted message 13. What message was sent?
- Anatoly's RSA public key is n=35 and e=11.
- Send Anatoly the message 12. What is the encrypted message?
- Help Anatoly by finding the additional information to decode your message.
- You receive the encrypted message 28. What message was sent?
- 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.
- If G is a group of order p2 (where p is prime) then prove that G has an element of order p.
- (Rotman 2.64, i-vi) True or false with reasons:
- 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.
- 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.)
- The inclusion that maps Z into R is a homomorphism of additive groups.
- The subgroup {0} of Z is isomorphic to the subgroup {(1)(2)(3)(4)(5)} of S5.
- Any two finite groups of the same order are isomorphic.
- If p is a prime, any two groups of order p are isomorphic.
- 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.)
- (Rotman 2.67.i) Prove that the composite of homomorphisms is itself a homomorphism.
Part 2
- (Rotman 2.67.iv) Prove that two groups that are each isomorphic to a third group are isomorphic to each other.
- (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.
- Prove that a group homomorphism f from G to H is injective exactly when Ker(f) is just the identity.
- Prove that a group homomorphism f from G to H is surjective exactly when Im(f) is all of H.
- 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).
- 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.
- 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 1: Due Tuesday 3/6
- Part 2: Due Friday 3/9