Homework for Week 6

Part 1

  1. List all cosets of H in G, when
    1. H = <2>, G = integers mod 8
    2. H = <(123)>, G = S3
    3. H = <(12)(3)>, G = S3
  2. (Rotman 2.52vi) True or false with reasons: If H is a subgroup of G, then the intersection of two (left) cosets of H is a (left) coset of H.
  3. (Rotman 2.63)
    • Show that a left coset of <(12)(3)> in S3 may not be equal to a right coset of <(12)(3)> in S3; that is, find w in S3 so that w<(12)(3)> is not the same as <(12)(3)>w.
    • Let G be a finite group and let H be a subgroup of G. Prove that the number of left cosets of H in G is the same as the number of right cosets of H in G.
  4. If H and K are subgroups of a group G and if |H| and |K| are relatively prime, then prove that the intersection of H and K consists just of the identity element.
  5. What is the largest order of an element in Sn for n=1,2,3,...,10? What is an upper bound on the order of a subgroup of Sn for n=1,2,3,...,10?
  6. (Rotman 2.46) If G is a group with an even number of elements, prove that the number of elements in G of order 2 is odd. In particular, G must have some element of order 2.

Part 2

  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.