Abstract Algebra — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

All parts of a question are to be attempted together. Each part carries 7.5 marks. 1. (a) (b) (c) Define the group GL(2,F), where F is any field. In GL(2, Z13), find the inverse of . Prove that a group G is Abelian iff (ab)–1 = a–1b–1 ∀ a, b ∈ G. Define the group U(n), for n ∈ N Construct the Cayley table for U(20). Find the inverse of each element of U(20). 2. (a) (b) (c) Let G be an Abelian group under multiplication with identity e. Prove that the subset H = {x2 | x ∈ G} is a subgroup of G. Let G be a group. Define the Centralizer, C(a), of an element a in G. Prove that C(a) = C(a–1) ∀ a ∈ G. Define the order of an element in a group G. Find the order of each element in the group U(15). 3. Define a cyclic group. Show that U(14) = <3> = <5>. Is U(14) = <11>? (b) Define the order of a permutation. Determine whether the following permutations are even or odd : (c) State & prove Lagrange’s theorem for groups. 4. (a) (a) Prove that a quotient group of a cyclic group is cyclic. (b) Define a normal subgroup of a group. Let (c) Is H a normal subgroup of the group GL(2, R)? Show that map φ: R* → R* defined by φ(x) = |x| is a group homomorphism. Also show that Ker φ is a cyclic group. 5. (a) (b) (c) 6. (a) (b) (c) (i) Prove that the set is a subring of the ring R of 2 × 2 matrices over Z. (ii) Prove that the ring Z p of integers modulo p, where p is a prime number, is an integral domain. Prove that intersection of any collection of subrings of a ring R is a subring of R. Does the result hold for union of subrings ? Justify. Show that the set unity. is a commutative ring with Define an ideal of a ring. State the ideal test. Hence prove that nZ is an ideal of Z, where n ∈ Z. Let f: R → S be a ring homomorphisms. Prove that (ii) f(B) = {r ∈ R : f(r) ∈ B] is an ideal of R, where B is an ideal of S. Prove that the only ideals of a field F are {0} and F itself.