Dept of Mathematics Statistics: Algebra Prelims Department of Mathematics Statistics August, 2005 setlengthheadheight0bp setlengthheadsep0bp setlengthtopmargin36bp setlengthtextheight704bp Directions: There are 8 questions, all of the same weight Please take the time to ensure accuracy and completeness, especially for the questions you find easiest (Completeness does not mean excessive verbosity You should not attempt to prove standard propositions that you cite except where the proof of a standard proposition is explicitly sought.) The ring of integers will be denoted by Z and its field of fractions by Q Prove that a nonabelian group of order 2p, p an odd prime, must have a trivial center When F is a field, let GL_n(F) denote the group of all invertible n n matrices in F under the operation of matrix multiplication, and let SL_n(F) denote its subgroup defined by restricting to matrices of determinant 1 Find a subgroup H of GL_n(F) such that GL_n(F) is isomorphic to the semidirect product of H with SL_n(F) Prove that the number of elements in any finite field must be a prime power Let ZmZ denote the ring of integers modulo m Let r,s be positive integers What element of Z generates the ideal rZ sZ What is the kernel of the canonical ring homomorphism ZrsZ ZrZ ZsZ Find an integer t such that ZrZ ZsZ ZtZ Let M be a 3 3 matrix over the rational field Q whose characteristic polynomial is t³ 2t² 4t 8 Find: all possible sequences of (polynomial) invariant factors for M representatives of the different possible similarity classes of such matrices M For any integer n 3 let D_n denote the nth dihedral group, i.e., the group of order 2n that is the semidirect product of the cylic group ZnZ with Z2Z for the unique nontrivial action (by automorphisms) of the latter on the former, or, equivalently, the group of symmetries of a regular ngon Describe D_n by generators and relations Show that every automorphism of the dihedral group D₃ is inner, i.e., is the conjugation by some element of D₃ Show that for any n odd, n 5, the dihedral group D_n has an automorphism that is not inner For any integer n 1, explain how to find a field K and a polynomial f(x) K [x] of degree n so that L K [x](f(x)) is a Galois field extension of K with cyclic Galois group of order n Let F_p denote the field ZpZ of p elements In M_n(F_p), the ring of n n matrices with entries in F_p, let N be the n n Jordan block matrix with 1's on the first superdiagonal (and 0's everywhere else including the main diagonal) Let F_p [N] be the F_palgebra of polynomials with coefficients in F_p evaluated at N, let U be the group of units in F_p [N] Describe the elements of U Find the exponent of U when p n Describe the isomorphism type of U as a finite abelian group when p n