2009 Honors Examination in Algebra

[1](a) Let H be a subgroup of a finite group G. Show that the number of conjugates gHg^-1 of H in G divides the index [G: H] of H in G.

(b) Now let G act on the left on a set X, and consider the orbit O_x = {g_x | g ∈ G} of an element x ∈ X. Let G_x = {g ∈ G | g_x = x} denote the stabilizer (isotropy subgroup) of x. Are all of the subgroups G_y for y ∈ O_x conjugate to G_x? Does this list include every subgroup conjugate to G_x? Does each distinct subgroup appear the same number of times?

(c) A recurring theme in group theory is that we need look no further than the group G itself, to understand questions like the above. Can you construct a set Y from G itself which admits a left action by G, and has an orbit with the same structure as O_x above?

[2](a) Classify the finite groups of order 12.

(b) Two groups are the nonsingular matrices of the form A with integer entries mod 2, and the nonsingular matrices of the form B with integer entries mod 3, where

Which groups are they?

[3](a) Construct explicitly the finite field F₂₅ of order 25, as a quotient F₅[x]/(f(x)) for a polynomial f(x).

(b) In your notation, what is the irreducible polynomial over F₅ of the element x + 1 of F₂₅?

(c) Let G be the group of invertible 2 × 2 matrices with entries in F₂₅. What is the order of G? How many conjugacy classes of G consist of elements of order 3? Find a representative for each of these conjugacy classes.

(d) Find a 3-Sylow subgroup of G. How many 3-Sylow subgroups does G have?

[4](a) Let G be a finite group acting on a finite set X. Prove Burnside's formula: The number of orbits of this action is

1/|G| Σ_g∈G|X^g|

where X^g denotes the set of elements in X that are fixed by g.

(b) Consider an unshaded n × n checkerboard, like the board shown for n = 5:

We would like to count the number of ways of placing two checkers on such a board, up to symmetry, where the board may be flipped or rotated. How would you formulate this problem as a group action? What is the group G? What is the set X? How many ways are there?

(c) Identify opposite sides of the board to form a torus, which may be flipped, rotated, or translated. Now what is the group G? How many ways can we place two checkers on this torus, up to symmetry?

[5](a) Describe the irreducible representations of the symmetric group S₃, and their character table.

(b) How many irreducible representations does the symmetric group S₄ have?

(c) How many irreducible representations does the alternating group A₅ have?

[6](a) Let A be a square matrix that satisfies the matrix equation A² - 5A + 6I = 0, where I is the identity matrix. What is the structure of the ring Z[A] generated by A over Z?

(b) Find a formula for Aⁿ, as an element of Z[A].

[7](a) Define irreducible and prime for an integral domain. What is the relationship between euclidean, unique factorization, and principal ideal domains?

(b) List a few of the smallest irreducible elements of the integral domain

Z[√-2] = { a + b√-2 | a, b ∈ Z }

as measured by the norm N(a + b√-2) = a² + 2b². (In the complex numbers, this is length squared.)

(c) Is Z[√-2] a unique factorization domain?

(d) More generally, consider the integral domain

Z[√-c] = { a + b√-c | a, b ∈ Z }

where c is a positive integer. For which of these domains does the integer 6 factor uniquely into irreducibles?

[8] Let C[s, t] be the ring of polynomials on C², and let C[w, x, y, z] be the ring of polynomials on C⁴. In other words, each element f of C[s, t] is a polynomial function f: C² → C, and each element g of C[w, x, y, z] is a polynomial function g : C⁴ → C.

Consider the map α: C² → C⁴ given by α(s, t) = (s³, s²t, st², t³). The map α induces a ring homomorphism β: C[w, x, y, z] → C[s, t] given by the rule g |→ g o α. In other words, we restrict polynomials on C⁴ to the image of α, and pull back by α to obtain polynomials on C².

(a) Describe the image of β as a subring R ⊂ C[s, t].

(b) Describe the kernel of β as an ideal I ⊂ C[w, x, y, z].

(c) An ideal P in a commutative ring R is prime if and only if the corresponding quotient ring R/P is an integral domain. Show that I is a prime ideal in C[w, x, y, z]. Find a strictly ascending chain I ⊂ J ⊂ K of prime ideals in C[w, x, y, z], where K is a maximal ideal.

[9] For any cubic polynomial f(x) = (x - α)(x - β)(x - γ), the discriminant D of f is defined to be

D = (α - β)²(α - γ)²(β - γ)²

(a) Derive a formula for D in terms of p and q, for polynomials of the form f(x) = x³ + px + q.

(b) Prove that the discriminant of an irreducible cubic polynomial f(x) in Q[x] is a square in Q if and only if the degree of its splitting field is 3.

(c) What is the degree of the splitting field of f(x) = x³ - 7x + 7?

[10](a) Prove the Eisenstein Criterion: If f(x) ∈ Z[x] and p is a prime, such that the leading coefficient of f(x) is not divisible by p, every other coefficient of f(x) is divisible by p, but the constant term is not divisible by p², then f(x) is irreducible in Q[x].

(b) Show that x⁵ - 81x + 3 is irreducible in Q[x].

(c) Can x⁵ - 81x + 3 = 0 be solved by radicals?