2007 Honors Examination in Algebra-
1. Let Sn 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 Sn (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, . . . , xn} 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 gi ∈ G such that gix1 = xi).
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 + · · · + anxn|ai ∈ 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 → Sn.)
(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 Sn 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, . . . , Cl. Let CG = {∑g∈G λgg|λg ∈ C} denote the group algebra of G over the complex numbers C.
(a) Show that the class sums Ci- = ∑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 φz(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(vi) = vσ(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 Sn.