Abstract Algebra — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

Non-Program : Major/Minor - DSC 1. (a) Define a group and show that the set of integers under multiplication is not a (7.5) (b) Show that G = {6 g) : a # E ER) is an abelian group under (7.53) (c) Find the inverse of y 11 € GL(2, 713). 2. (a) Define the order of an element "a" in G. Also find the orders of every element (7.5) (b) Define center of a group and centralizer of a EG. Prove that C(a) = C(ar-). (e) Prove that in any group an element and its inverse will have the same order. (7.5) 3. (a) Leta = 3451688l and = ll 33465241 Write a, B and aß as products of disjoint cycles and as products of 2-cycles. (b) List the elements of the subgroups (3) and (7) in U(20). (7.5) (C) Let H = {0, In, ‡2n, ‡3n,...}. Find all left cosets of H in Z. (7.5) 4. (a) Let H be a normal subgroup of a group G and K be any subgroup of G, then (7.5) prove that HK = {hk] h € H, k € K} is a subgroup of G (* Wang d.15 D, belate Caley table for Da, the group of symmeties of a (15) 5. (a) State the subring test. Prove or disprove that the set is a subring of the ring of all 2 × 2 matrices over Z. (b) Define characteristic of a ring. Let R be a ring with unity 1. Prove that if 1 has (7.5) infinite order under addition then charR = 0. If / has order n under addition, then charR = n. c) Prove that a finite Integral Domain is a field. Hence, show that Zap is a field, (7.5 here p is a prime number 6. (a) Let f:R→S be a ring homomorphism. Prove that (7.5) (i) Kerf is an ideal of R. Every ideal A of a ring R is the kernel of a ring homomorphism of R. (b) Prove that if an ideal I of a ring R contains a unit, then I = R. Hence, prove (7.5) that the only ideals of a field Fare {0} and F'itself. (c) State and prove the first Isomorphism Theorem for Rings. (7.5)