Homework for Week 6
Part 1
- List all cosets of H in G, when
- H = <2>, G = integers mod 8
- H = <(123)>, G = S3
- H = <(12)(3)>, G = S3
- (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.
- (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.
- 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.
- 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?
- (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
- 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 1: Due Tuesday 3/1
- Part 2: Due Friday 3/4