Reference no: EM131104203

Honors Examination: Algebra

1. Let O(n) denote the orthogonal group (that is, the set of all real n×n matrices A such that AA^T = I), and let SO(n) denote the subgroup consisting of matrices with determinant 1.

(a) Prove that, for all n, SO(n) is a normal subgroup of O(n) of index 2.

(b) For which n (if any) is O(n) isomorphic to SO(n) × {±1}? Explain your answer carefully.

2. Let X denote the set of 2 × 2 complex matrices, and let G = GL₂(C).

(a) If G acts on X by left-multiplication, how does X decompose into orbits?

(b) If G acts on X by conjugation, how does X decompose into orbits?

(c) For each of the orbits in (a) and (b), pick a representative element and describe its stabilizer.

3. Let F_q denote the field with q elements, and let SL_n(F_q) denote the group of n × n matrices over F_q with determinant 1.

(a) Show that |SL₂(F₃)| = 24.

(b) Is SL₂(F₃) isomorphic to the symmetric group S₄? Explain your answer carefully.

(c) Let G be the group of rotations of a cube. Is G isomorphic to S4? Explain your answer carefully.

4. Let M(2) be the group of isometries of R². Consider the subgroup O(2) consisting of rotations and reflections fixing the origin, and the subgroup T(2) consisting of translations.

(a) Determine whether O(2) and T(2) (or perhaps both) are normal subgroups of M(2).

(b) Suppose that H is a subgroup of M(2) containing rotations about two distinct points. Prove that H contains a nontrivial translation. Hint: one approach (among several) is to look at commutators.

5. Let G be a finite group.

(a) Define the regular representation of G over the complex numbers. What is its character χ_R?

(b) How does χ_R decompose into irreducible characters?

(c) Show that if Ψ is a character of G and Ψ(g) = 0 for all g ≠ 1 in G, then Ψ is an integral multiple of χ_R.

6. A famous theorem states that every finite abelian group G is isomorphic to a direct product of cyclic groups of prime power order, and that this representation is unique up to permutation of the factors. Prove the second part of this statement, i.e., uniqueness of the representation.

7. Let R = Z[x], the ring of polynomials in x with integer coefficients, and let I = (3, x) be the ideal generated by 3 and x.

(a) Is I principal? Is it prime? Is it maximal? Explain your answers.

(b) Answer the questions in (a) for J = (x² + 1), the ideal generated by x² + 1.

8. Let G be a finite, nonabelian simple group, and consider representations of G over the complex numbers.

(a) Show that there cannot exist more than one linear (i.e., degree 1) character.

(b) Show that there cannot exist any irreducible characters of degree 2.

9. Let G be a finite group, and let p be the smallest prime dividing |G|. Prove that any subgroup H ⊆ G of index p is normal in G.

10. Let ω = e^2πi/n and let F_ω = Q(ω).

(a) What is [F_ω : Q]?

(b) Let G = Gal(F_ω, Q) be the Galois group of F_ω over Q. Describe G and compute its order.

(c) List all of the subfields of F_ω when n = 8, and make a diagram showing their containment relationships.

(d) (If you have time) For which n is G a cyclic group? Discuss, giving arguments and/or examples to support your answer.