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.

THE END