Honors Examination Algebra (version A) Spring 2004-

1. Let A and B be subgroups of a group G with A ∩ B = {1}.

(a) Show that if A and B are normal subgroups, then ab = ba for all a ∈ A and b ∈ B.

(b) Show that if ab = ba for all a ∈ A and b ∈ B, then AB = {ab | a ∈ A, b ∈ B} is a subgroup of G and AB ≅ A × B.

2. Exhibit all the permutations in S₇ that commute with α = (1 2)(3 4 5). Justify your answer.

3. An automorphism of a group G is an isomorphism from G to itself. A subgroup H of G is called characteristic, denoted Hchar G, if every automorphism of G maps H to itself (that is, φ(H) = H for all automorphisms φ of G).

(a) Prove that characteristic subgroups are normal.

(b) Prove that: if Kchar H and H / G, then K / G (here / denotes normal subgroup).

(c) Give an example of a normal subgroup that is not characteristic.

4. A group is simple if it has no proper nontrivial normal subgroups. This problem will prove that there is no simple group G of order 45. By way of contradiction, suppose that G is a simple group of order 45 (keep in mind that somewhere below you need to use the assumption that G is simple).

(a) The Sylow theorems tell us that G has a 3-subgroup P of order 9. There are 5 cosets of P. Let's call them {g₁P, g₂P, g₃P, g₄P, g₅P}. Show that left multiplication by an element g ∈ G gives a permutation of these cosets.

(b) Part (a) allows us to associate each element of G with a permutation in S5. Thus, it gives us a map φ: G → S₅. Show that the map φ is injective (i.e., one-to-one).

(c) Argue that S₅ does not have a subgroup of order 45, and thus G cannot be simple.

5. True/False? Justify Your answers

(a) 4x² + 6x + 3 is a unit in Z₈[x].

(b) Z₇[√3] is a field.

(c) (x, y) is a maximal ideal in Z[x, y]

(Notation: Z[x, y] is the ring of polynomials in two variables x and y with integer coefficients, and (x, y) is the ideal generated by x and y).

6. Let R be a commutative ring with 1 and let s ∈ R. Define the annhilator of s to be Ann(s) = {a ∈ R | sa = 0}.

(a) Prove that Ann(s) is an ideal of R.

(b) Describe Ann(s) when s is a unit.

(c) If e ∈ R satisfies e² = e, then show that Ann(e) = (1 - e)R.

(d) It is tempting to think that s + Ann(s) is not a zero divisor in the quotient ring R/Ann(s) (since we are dividing out all the stuff that sends s to 0). Find a counterexample to this statement in Z₁₂.

7. Let R be a commutative ring with 1 and let a ∈ R. Let R' = R[x]/(ax - 1).

(a) Describe R' in the case where a is a unit.

(b) Describe R'in the case where a is nilpotent, i.e., aⁿ = 0 for n ∈ Z^>0.

8. Let f(x), g(x) ∈ Q[x] be irreducible polynomials with a common zero z ∈ C. Prove that they generate the same principal ideal (f) = (g). (Hint: think about the ideal (f, g) generated by them both).

9. Let F be a finite field with q elements and let a be a nonzero element of F. Prove that if n divides q - 1, then xⁿ - a has either no solutions in F or has n distinct solutions in F.

10. Let F₄ = {0, 1, α, α²} be a field of order 4. Let G be the group of invertible 2 × 2 matriceswith entries a, b, c, d ∈ F₄, whose column sums are 1 (i.e., a + c = b + d = 1; these are called stochastic matrices).

(a) Give the multiplication and addition table for F₄.

(b) Show that G is a nonabelian group of order 12.

(c) Up to isomorphism, there are 3 nonabelian groups of order 12: the dihedral group D₆, the alternating group A₄, and Q₆ = h s, t | s⁶ = 1, s³ = t², sts = t i. Which group is it? (You do not need to exhibit an isomorphism).