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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 ?