Mathematics 243, section 3 – Algebraic Structures Problem Set 9 due: Friday, November 30

‘A’ Section

1. For each of the following values of n,

• Find all distinct generators of the group (ℤ_{n}, +),

• Find all subgroups of (ℤ_{n}, +) and their orders

• Find all elements of (ℤ_{n}^{×}, ·) and their orders (for the multiplication operation mod n now)

n = 13, 16, 30

(Use the “big theorem” on cyclic groups for as much of this as possible. It is not necessary to do a lot of computations in most cases.)

2. Let φ : ℤ_{18} → ℤ_{9} be defined by φ([x]) = [3x].

(a) Verify that φ is a group homomorphism.

(b) Determine the kernel of φ.

(c) Determine the image of φ.

‘B’ Section

1. Let G be a group and consider the mapping φ : G → G defined by φ(x) = x^{−1}. Show that φ is always one-to-one and onto, but that φ is an isomorphism of groups if and only if G is an abelian group.

2. An automorphism of a group G is an isomorphism of groups φ : G → G (that is, the domain and the range are both the same group G).

(a) Let A = {a, b, c} and G = S(A) be the group of permutations of A. Show that φ : G → G defined by φ(f) = R_{a} ∘ f ∘ R_{a} is an automorphism of G.

(b) Show that the collection of all automorphisms of a general group G is itself a group under the operation of function composition.

(c) Show if G is a general group and g ∈ G, then the conjugation mapping defined by φ_{g}(x) = gxg^{−1} is an automorphism of G. (Note that the example in part (a) has this form.)

(d) Show that the collection of φ_{g} for all g ∈ G (as in part (c)) is a subgroup of the group of automorphisms of G.

3. We can consider isomorphism of groups as a relation on the collection of all groups: GRH ⇔ there exists an isomorphism φ : G → H. Show that isomorphism of groups is an equivalence relation on the collection of all groups.

4. Let G = ⟨a⟩ be a cyclic group and let φ : G → H be a group homomorphism. Show that if we know the one element φ(a), then we know where φ maps every element of G.