Math 171 abstract algebra fall 2014- sample midterm

Question

Math 171: Abstract Algebra, Fall 2014- Sample Midterm

1. Let X be the set of polynomials in the variables x₁, x₂, x₃, x₄, x₅ with coefficients in Z. Let S₅ act on X as follows:

σ · f(x₁, x₂, x₃, x₄, x₅) = f(x_σ(1), x_σ(2), x_σ(3), x_σ(4), x_σ(5)).

For instance, if f = x₁²x₂ + x₂²x₃⁸x₄ + x₅³, and σ = (1 2 3)(4 5), then

σ · f = x²₂x₃ + x²₃x⁸₁x₅ + x³₄.

Consider the polynomial f = x₁ + x₂ + x₃ + x₄⁴ + x⁴₅.

(a) Determine |Gf|, the orbit of f. Justify.

(b) Determine |Gf|, the stabilizer of f. Justify

2. Suppose G and H are abelian groups of order 27, and for every positive integer k the number of elements of order k in G equals the number of elements of order k in H. Prove G ≅ H.

3. Let G be a group. Define Aut(G) to be the set of isomorphisms from G to itself. Aut(G) forms a group under composition. For any h ∈ G, define φ_h: G → G to be the homorphism from G to itself given by

φ_h(g) = hgh^-1.

Observe that for any h ∈ G, φ_h is in Aut(G).

(a) Show that Inn(G) := {φh | h ∈ G} is a normal subgroup of Aut(G). You may assume without proof that Inn(G) is a subgroup of Aut(G), and that the homomorphism φ_e is the identity of Aut(G).

(b) Prove Inn(G) ≅ G/Z(G). (Hint: Consider the homomorphism ψ: G → Inn(G) given by ψ(g) = φ_g.)

4. Let G be a non-abelian group of order 39.

(a) Prove |Z(G)| = 1.

(b) Use the Class Equation to prove G has exactly 4 conjugacy classes of size 3 and 2 conjugacy classes of size 13.

5. With help from the Sylow Theorem, prove that if G is a group of order 65 then G is cyclic.