# MATHEMATICS MODEL PAPER THIRD SEMESTER - ABSTRACT ALGEBRA commom for B.A and BSC 2017

 MATHEMATICS MODEL PAPER  THIRD SEMESTER - ABSTRACT ALGEBRA  commom for B.A and BSC 2017 SECTION – A I. Answer any five questions. Each question carries five marks. 5 x 5 = 25 M 1. Show that a group G is abelian if and only if (ab)2 = a2b2 " a, b ÎG. 2. Define order of an element in a group and prove that in a group G, O(a) = O(a–1) for a Î G. 3. Prove that intersection of two sub groups of a group G is a sub group of G. 4 Show that a sub group H of a group G is normal iff xHx–1 = H"x ÎG. 5. Show that the mapping f: G ® G such that f(a) = a–1 " a Î G is an automorphism of a group G iff G is abelian. 6. If f is a homomorphism of a group G in to a group G’ then prove that kernel of f is a normal sub group of G. 7. Express the product (2 5 4) (1 4 3) (2 1) as a product of disjoint cycles and find its inverse. 8. Prove that an infinite cyclic group has exactly two generators. SECTION – B II. Answer the all five questions. Each carries ten marks 5 x 10 = 50 M 9. a) Show that a finite semi group satisfying cancellation jaws is a group. (OR) b) Show that the set G of rational numbers other than 1 is an abelian group with respect to the operation Å defined by a Å b = a + b – ab "a, b ÎG. 10. a) Prove that the necessary and sufficient condition for a non empty sub set H of a group G to be a sub group is that a, b Î H Þ ab–1ÎH. (OR) b) State and prove Lagrange’s theorem. 11. a) If H is a normal sub group of a group G then prove that the set g/H of all cosets of H in G is a group with respect to coset multiplication. (OR) b) Prove that a subgroup H of a group G is a normal subgroup of G iff the product of two right cosets of H in G is again a right coset of H in G. 12. a) If f is a homomorphism of a group G in to a group G’ then prove that f is an into isomorphism iff ker f = {e} (OR) b) State and prove fundamental theorem on homomorphism. 13. a) State and prove Cayley’s theorem. (OR) b) Prove that every sub group of a cyclic group is cyclic.

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