2018 Fall Algebra

Problem 1.

Let be a group of size 42.

(a) Prove that has a subgroup of order 6 and any two such subgroups are conjugate in .

(b) Deduce that , where is a normal subgroup of order 7.

Problem 2.

Let be a prime. Show that for any Sylow -subgroup there exists a basis in the vector space such that consists of -linear maps given, in that basis, by an upper-triangular matrix with 1 on the diagonal.

Problem 3.

For a group , let and let . We say is nilpotent if for some . Prove that if is a -group, i.e. for some prime , then is nilpotent.

Problem 4.

Let be the subring in the field of rational numbers, given by the fractions with and with . Describe the ideals of . Is a PID?

Problem 5.

Let be the ring of Gaussian integers.

(a) Show that is a Euclidean Domain.

(b) Find a decomposition of into a product of irreducibles in .

(c) Find a decomposition of into a product of irreducibles in .

Problem 6.

Let be a nonzero finite-dimensional vector space over the complex numbers.

(a) If and are commuting linear operators on , prove that each eigenspace of is mapped into itself by .

(b) Let be finitely many linear operators on that commute pairwise. Prove that they have a common eigenvector in .

(c) If has dimension , show that these exists a nested sequence of subspaces

where each has dimension and is mapped into itself by each of the operators .

Problem 7.

Let be an matrix with complex coefficients and assume that every eigenvalue of satisfies . Consider the matrix

Find the invariant factors of in terms of the invariant factors of and prove that is similar to a real valued matrix.

Problem 8.

Find the Galois group of over and .

Problem 9.

Let be a finite Galois algebraic extension with no proper intermediate fields. Prove that is prime.

Problem 10.

Let denote the quaternion group, i.e.

with , , and .

(a) Classify the conjugacy classes of .

(b) Constrcut the character table of .