2007 Honors Examination in Algebra-

1. Let S_n denote the symmetric group of permutations of {1, 2, . . . , n}. Consider S₃ ⊆ S₄ in the natural way such that the permutations in S₃ fix the number 4.

(a) Find a set of coset representatives of S₃ in S₄ (i.e., one from each coset).

(b) Find a property on the permutations of S₄ such that σ and τ share this property if and only if they are in the same coset.

(c) Generalize both (a) and (b) to S_n (with proof).

2. Prove from basic principles that in a commutative ring R with unity 1, an ideal M is maximal if and only if R/M is a field.

3. Show that x³ - x - 1 and x³ - x + 1 are irreducible over Z₃ (the integers mod 3). Construct their splitting fields and explicitly exhibit an isomorphism between these splitting fields.

4. Let G be a group with subgroup H. For g ∈ G define the (H, H)-double coset of g to be

HgH = {h₁gh₂|h₁,h₂∈H}.

(a) Show that the (H, H) double cosets partition G but that, unlike left cosets, different (H, H)-double cosets need not have the same cardinality.

(b) Suppose that G acts transitively on the set X = {x₁, . . . , x_n} and that H = {g ∈ G|gx₁ = x₁}. Show that the number of orbits of X under the action of H is the same as the number of (H, H)-double cosets. (Hint: since G acts transitively, there exist elements g_i ∈ G such that g_ix₁ = x_i).

5. True or False? Justify your answers.

(a) The principal ideal (1 + √-5) is maximal in Z[√-5].

(b) Z[i]/(2 - i) ≅ Z₅, (Here, Z[i] denotes the ring of Gaussian integers).

(c) The ring S ⊆ Z[x] consisting of polynomials with zero coefficient on the linear term

S = {a₀ + a₂x² + a₃x³ + · · · + a_nxⁿ|a_i ∈ Z}

is a unique factorization domain (hint: think about x² and x³).

6. Let E be the splitting field of the polynomial x³ - 2 over Q.

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

(b) Find a primitive element α such that E = Q(α). (Note that it suffices to show that α is not contained in any proper subfield of E.)

7. (a) Let G be a group and H ≤ G a subgroup with [G : H] = n. Prove that there exists a normal subgroup N $ G such that

(a) [G : N] divides n! and (b) N ⊆ H

(Hint: Let G act on the set of left cosets X = {gH|g ∈ G} to get φ : G → S_n.)

(b) Use the previous result to show that any group of order 24 must have a normal subgroup of order 4 or 8. (Hint: let H be a Sylow-2 subgroup.)

8. Let R be a commutative ring with unity 1. An element t in an R-module M is called a torsion element if rt = 0 for some nonzero element r ∈ R. The set of torsion elements in M is denoted tor(M).

(a) Prove that if R is an integral domain then tor(M) is a submodule of M and show that M/tor(M) has no nonzero torsion elements (i.e., it is torsion free).

(b) Give an example of a ring R and an R-module M such that tor(M) is not a submodule.

(c) Show that if R has zero divisors, then every nonzero R-module has torsion elements.

9. Every permutation in S_n is either even or odd. Define the sign of a permutation σ to be sign(σ) = 1, if σ is even, and sign(σ) = -1, if σ is odd. Prove that

sign(σ) = ∏_{1≤iσ(i) - σ(j)/i-j.}

10. Let G be a finite group with conjugacy classes C₁, C₂, . . . , C_l. Let C_G = {∑_g∈G λ_gg|λ_g ∈ C} denote the group algebra of G over the complex numbers C.

(a) Show that the class sums C_i^- = ∑_c∈Ci c are in the center Z(CG).

(b) Let V be a G-module over C. For z ∈ Z(CG), show that the map φz : V → V given by φ_z(v) = zv is a G-module homomorphism.

(c) Suppose that V is an irreducible G-module. What does Schur's lemma say about the homomorphism φz? Use this to describe multiplication of vectors in V by an element z of the center.

(d) As an example, let V be the permutation module of S₄ spanned by v₁, v₂, v₃, v₄ with σ(v_i) = v_σ(i). Let W = span(v₂ - v₁, v₃ - v₁, v₄ - v₁). Then W is an irreducible submodule of V (you do not need to prove this). Let C be the conjugacy class consisting of two-cycles. Describe the action of C^- on W. If possible, generalize to S_n.