January 2016 Qualifying Examination If you have any difficulty with the wording of the following problems please contact the supervisor immediately. All persons responsible for these problems, in principle, will be accessible during the entire duration of the exam. This exam has two parts - Abstract Algebra and Linear Algebra. Please separate your answers for the two parts and staple them separately. 1
Abstract Algebra In what follows, n is a positive integer, and p is a prime number; 1 G stands for the identity in a multiplicative group G, and {1 G } is called the trivial subgroup of G; the symbols H < G (respectively, H G) denote that H is a subgroup (respectively, normal subgroup) of G; the symbols [G : H] stand for the index of H in G; if a G, the symbols G (respectively, a ) stand for the order of the group G (respectively, the order of a in G); the symbols S n (respectively, A n ) stand for the permutation (respectively, alternating) group on n letters. 1. Let H < G. Let C := {gh g G}, and let Γ be the set of permutations γ a on C defined by γ a (gh) := agh. Let f : G Γ, be defined by f(a) := γ a. (i) (2 points) Prove that Γ is a group, and that f is a group homomorphism with Ker(f) < H. (ii) (2 points) Let [G : H] = n. Assume that H contains no non-trivial, normal subgroups of G. Prove that G is isomorphic to a subgroup of S n. (iii) (2 points) Let H = p, and G = pn, where p > n. Prove that H G. 2. Let G be a finite group. (i) (2 points) Prove that there is a natural number n such that G is isomorphic to a subgroup of A n. (ii) (2 points) Let a G. Then the conjugacy class of a is A a = {g 1 ag g G}. Suppose that G has precisely two conjugacy classes. Prove that G = 2. (iii) (3 points) Let H, K, G be groups such that G = H K with G cyclic, of order p n. Prove that H = 1 or K = 1. 3. Let R be a commutative ring with (multiplicative) identity 1 R. Let S be a multiplicatively closed subset of R, i.e., st S for all s, t S. Let J be a proper ideal of R, and Rad(J) be the intersection of all prime ideals P containing J. You may assume the fact that Rad(J) = {r R r n J for some n > 0} without proving it. (i) (2 points)let I be and ideal of R, maximal with respect to I S = (i.e., I is an ideal of R such that I S =, and whenever J is an ideal of R, containing I such that J S =, then I = J). Prove that I is a prime ideal of R. (ii) (3 points) Let I 1 and I 2 be proper ideals of R. Prove that Rad(I 1 I 2 ) = Rad(I 1 ) Rad(I 2 ). (iii) (3 points)let m 1 and m 2 be two distinct maximal ideals of R. Prove that m 1 m 2 = m 1 m 2. What is Rad(m 1 m 2 )?
4. Let F be the field with p n elements, and G := F {0 F } be the multiplicative (abelian) group of F with 1 G = 1 F. Note that G = p n 1. (i) (3 points) Prove that there is a positive integer t G such that a t = 1 G for all a G and such that a 0 = t for some a 0 G. (ii) (3 points) Prove that t = G, and hence that G is cyclic. (iii) (3 points) Let a 0 G be a generator for G, i.e., G =< a 0 >. Let f(x) Z pz [X] be the minimal polynomial for a 0 over Z, i.e., f(x) is monic, and pz irreducible in Z pz [X], and f(a 0) = 0 F. Prove that deg(f(x)) = n.
Linear Algebra All answers must be proven to receive credit. Suppose that V and W are finite dimensional vector spaces over a field F and φ : V W is a linear map. Suppose that β is an ordered basis of V and β is an ordered basis of W. The matrix M β β (φ) of φ with respect to the bases β and β is the unique matrix which satisfies M β β (φ)(v) β = (φ(v)) β for v V, where (v) β is the coordinate vector of v with respect to β and (φ(v)) β is the coordinate vector of φ(v) with respect to β. 1) (5 points) Show that for any two n n matrices A, B over a field F, the eigenvalues of AB are the same as the eigenvalues of BA. 2) (5 points) Suppose that A, B, C, D are four commuting n n matrices over a field F. Prove that ( ) A B det = det(ad BC). C D Left hand side is the determinant of a 2n 2n matrix and the right hand side is the determinant of the n n matrix. 3) (5 points) Suppose that A is a square matrix over Q such that its minimal polynomial is m(x) = (x 2) 3, its characteristic polynomial is χ(x) = (x 2) 7 and that dim Kernel L A 2I = 3, where L A 2I is the linear map determined by multiplication by A 2I. a) Up to permutation of Jordan blocks, how many distinct Jordan forms are possible for A? What are they? Show your work. b) Suppose that we have the additional information that dim Kernel L (A 2I) 2 = 6. In this case, how many distinct Jordan forms are possible for A, up to permutation of Jordan blocks? What are they? Show your work. 4) (5 points) Suppose that A is an m n matrix (m may be not equal to n). a) Suppose that A has real coefficients. Show that the n n matrix A t A is invertible if and only if rank A = n. b) Suppose that A has complex coefficients and rank A = n. Is it always true that A t A is invertible? If your answer is that A t A is invertible then give a proof. If your answer is that A t A is not always invertible then give an explicit counterexample.
5) (5 points) Suppose that V is a finite dimensional vector space over R. a) Suppose that <, > is a symmetric bilinear form on V (That is, < v, w >=< w, v > for v, w V and < v, w > is bilinear in v, w V ). Let V 0 be the subspace of V consisting of all vectors v V such that < v, w >= 0 for all w V. Let {v 1,..., v n } be an orthogonal basis of V for the form (< v i, v j >= 0 if i j). Show that the number of integers i such that < v i, v i >= 0 is equal to the dimension of V 0. b) Let A = 1 2 2 2 1 1 2 1 1 and define a symmetric bilinear form <, > on R 3 by < x, y >= x t Ay for x, y R 3 (define R 3 to be the set of 3 1 real column vectors). Find an orthogonal basis of R 3 for this form. 6) (5 points) Suppose that V is a finite dimensional vector space over a field F with a basis β = {v 1,..., v n }. Let V be the dual space of V (the vector space of linear maps from V to F ) with dual basis β = {v 1,..., v n} defined by v i (v j ) = δ ij. Suppose that φ : V V is a linear map. The dual map φ : V V is defined by φ (λ) = λ φ for λ V. a) Show that M β β (φ ) = (M β β (φ))t. b) Let Q be the vector space of polynomials of degree 2 in x (the degree of zero is ) with coefficients in F. Let β = {1, x, x 2 } be the standard basis of Q. Let φ : Q Q be the linear map defined by φ(a 0 + a 1 x + a 2 x 2 ) = (a 0 2a 1 ) + (a 2 a 1 )x + (a 2 3a 1 + a 0 )x 2 for a 0, a 1, a 2 F. Find a basis of Kernel φ, expressing its members as linear combinations of the dual basis β to β.