Dept of Mathematics Statistics: Algebra PrelimsDepartment of Mathematics StatisticsAugust, 2005setlengthheadheight0bpsetlengthheadsep0bpsetlengthtopmargin-36bpsetlengthtextheight704bpDirections: 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 GLnF denote the group of all invertible nn matrices in F under the operation of matrix multiplication, and let SLnF denote its subgroup defined by restricting to matrices of determinant 1 Find a subgroup H of GLnF such that GLnF is isomorphic to the semidirect product of H with SLnF 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 rZsZ What is the kernel of the canonical ring homomorphism ZrsZZrZZsZ Find an integer t such that ZrZZsZZtZ Let M be a 33 matrix over the rational field Q whose characteristic polynomial is t32t24t8 Find: all possible sequences of (polynomial) invariant factors for M representatives of the different possible similarity classes of such matrices M For any integer n3 let Dn 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 Dn 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, n5, the dihedral group Dn has an automorphism that is not inner For any integer n1, explain how to find a field K and a polynomial fxKx of degree n so that LKxfx is a Galois field extension of K with cyclic Galois group of order n Let Fp denote the field ZpZ of p elements In MnFp, the ring of nn matrices with entries in Fp, let N be the nn Jordan block matrix with 1's on the first superdiagonal (and 0's everywhere else including the main diagonal) Let FpN be the Fpalgebra of polynomials with coefficients in Fp evaluated at N, let U be the group of units in FpN Describe the elements of U Find the exponent of U when pn Describe the isomorphism type of U as a finite abelian group when pn