Math 171 abstract algebra fall 2014- assignment 4 determine

Question

Math 171: Abstract Algebra, Fall 2014- Assignment 4

1. Let G be a cyclic group of order 36. For how many positive integers k ∈ {0, 1, 2, . . . , 35} is the set map φ : G → G given by φ(g) = g^k, a homomorphism? For how many positive integers k ∈ {0, 1, 2, . . . , 35} is φ surjective?

2. Determine all positive integers n ≥ 2 for which the group (Z/2ⁿZ)^× is cyclic. Recall (Z/2ⁿZ)^× is the set {[a] | gcd (a, 2ⁿ) = 1}, with group operation ×.

3. Let G be a finite group and suppose H, K ≤ G. Define the set H · K := {hk|h ∈ H, k ∈ K} ⊆ G.

(a) Show that the relation on H × K given by (h, k) ∼ (h', k') if hk = h'k' is an equivalence relation, and use it to prove |H · K| = (|H||K|/|H ∩ K|).

(b) If G is abelian, prove H · K ≤ G.

(c) Conclude that every abelian group of order 6 is cyclic.

4. For each of the following groups G, you are given a subgroup H of G. List the set of right H-cosets in G, and in each case, verify |G| = |H| · [G : H].

(a) G = D₈, H = {1, r²}.

(b) G = A₄, H = V where V = {1,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)}

5. (a) Let G be a group with subgroups H and K with |H| = 21 and |K| = 39. Prove H ∩ K is a cyclic group.

(b) Let G be a group of order 35. Suppose G has exactly one subgroup of order 5 and one subgroup of order 7. Prove G is cyclic.