2012 Honors Examination in Algebra

Part A

1. Prove that the number of inner automorphisms of a group G equals the index of the center [G: Z(G)].

2. Let Z[i] = {a + bi| a, b ∈ Z } denote the Gaussian integers. Describe the quotient ring Z[i]/(3). What are its elements? What sort of algebraic structure is it?

3. Let V be a vector space and T: V → V be a linear transformation (i.e., a linear operator on V). Show that if T² = T, then V is the direct sum V = Im(T) ⊕ Ker(T) of the image and the kernel of T. Describe how to pick a nice basis for V relative to this direct sum, and give the matrix of T in that basis.

Part B

4. Let S_n denote the symmetric group on {1, 2, . . . , n} and view S_n-1 ⊆ S_n as the permutations which fix n.

(a) Is S_n-1 a normal subgroup? Justify.

(b) The elements in each coset have a distinguishing feature. What is it? You should be able to identify that two permutations are in the same coset by this feature.

(c) Let S_n act on these cosets by left multiplication: g · (hS_n-1) = (gh)S_n-1. This is a familiar action of the symmetric group. Describe it clearly.

5. Construct a finite field of order 9 as a quotient E = F[x]/(p(x)). Find a generator of the multiplicative group of E.

6. A proper ideal Q of a commutative ring R is called primary if whenever ab ∈ Q and a ∉ Q then bⁿ ∈ Q for some positive integer n.

(a) Give an example of an ideal in Z that is primary but not prime.

(b) Prove: An ideal Q of R is primary if and only if every zero divisor in R/Q is nilpotent. (An element a in a ring is nilpotent if aⁿ = 0 for some positive integer n).

(c) Use part (b) to show that (x, y²) is a primary ideal in Q[x, y].

7. Let A be an abelian normal subgroup of G and write G¯ = G/A. Show that G¯ acts on A by conjugation: g¯ · a = gag^-1 where g¯ = gA. Under what condition on G is this action faithful? (i.e., the map G¯ → Sym(A) is one-to-one, where Sym(A) is the group of permutations of A). Give an explicit example illustrating that this action is not well-defined if A is not abelian.

Part C

8. Let H be a subgroup of the finite group G with n = [G : H] and let X be the set of left cosets of H in G. Let G act by multiplication on X, i.e., g ·(xH) = (gx)H, and let φ: G → Sym(X) be the corresponding homomorphism to the permutation group of X.

(a) Show that ker(φ) is the largest normal subgroup of G that is contained in H.

(b) Show that if n! is not divisible by |G|, then H must contain a nontrivial normal subgroup of G.

(c) Let |G| = 108. Show that G must have a normal subgroup of order 9 or 27.

9. The Galois group of E = Q(i, 4√2) over Q is Gal(E/Q) ≅ D₄, the dihedral group of order 8.

(a) Give a basis for E as a vector space over Q.

(b) The dihedral group is presented by D₄ = (σ, τ | σ⁴ = 1, τ² = 1, τσ = σ^-1τ}. Describe the field automorphisms that correspond to σ and τ.

(c) Find the fixed field corresponding to the subgroup hσi and the fixed field corresponding to (τ).

(d) Show that (τ) is conjugate to (σ²τ) and then illustrate how to use this information to find the fixed field of (σ²τ).

10. Let G = {g₁, g₂, . . . , g_n} be a finite group with |G| = n. As is done with the regular representation of G, label an ordered basis of Cⁿ with the elements of the group {eg₁, eg₂, . . . , eg_n}. Only now, act on the basis by conjugation instead of left multiplication. Thus

ρ : G → GL_n(C)

g |→ ρ_g

where ρ_g(e_h) = e_ghg^-1 .

(a) Prove that this is a representation.

(b) Discuss whether this representation is faithful (sometimes, always, never, when?).

(c) Discuss whether this representation is irreducible.

(d) Find a nice formula for the corresponding character χ(g).