Algebra Qualifying Exam
September 18, 2013
Do all five problems.
1. Let G be a group with subgroup H. Define N =
\
g −1 Hg.
g∈G
(a) Prove that N ⊆ H and that N is a normal subgroup of G.
(b) Prove that N is the largest normal subgroup of G contained inside
H. In other words, prove that if K is any normal subgroup of G with
K ⊆ H, then K ⊆ N .
2. Let G be a group with a normal subgroup N of order 2. If G/N is cyclic
prove that G is abelian.
3. Let F be a field. Prove that there is a ring isomorphism
φ : F [x]/(x2 − x) → F × F
(here (x2 −x) is the ideal generated by x2 −x). You must explicitly describe
φ.
4. Let V be a finite dimensional vector space over C, and let S and T be two
linear transformations from V to V . Assume that ST = T S and that the
characteristic polynomial of S has distinct roots.
(a) Show that every eigenvector of S is an eigenvector of T .
(b) If T is nilpotent, show that T = 0.
5. Let a, b ∈ R and let T : R3 → R3 be the linear transformation which is
orthogonal projection onto the plane z = ax + by (with respect to the usual
Euclidean inner-product on R3 ).
(a) Find the eigenvalues of T and for each find a basis for the corresponding eigenspace.
(b) Is T diagonalizable? Justify.
(c) What is the characteristic polynomial of T ?
(d) What is the minimal polynomial of T ?