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Posted Date: 25 Oct 2009 Posted By: rajshaker Member Level: Bronze
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2008 Andhra University M.Sc Mathematics ALGEBRA Question paper
M.Sc. DEGREE EXAMINATION, 2007 ( MATHEMATICS )
( FIRST YEAR ) ( PAPER - I )
510. ALGEBRA – I (New Regulations)
May ] [ Time : 3 Hours Maximum : 100Marks
PART A (8 × 5 = 40) Answer any EIGHT questions.
Each questions carries FIVE marks.
1. If H is a non-empty finite sub set of a group G and if H is closed under multiplication, prove that H
is a subgroup of G.
2. If G is a finite group and H is a subgroup of G, prove that O(H) is a divisor of O(G).
3. Prove that every permutation is the product of its cycles.
4. Define conjugate elements in a group G. Prove that conjugacy is an equivalence relation on
G.
5. Prove that a finite integral domain is a field.
6. State and prove the division algorithm for a polynomial ring.
7. Define ?linear span?. If S is a non-empty subse of a vector space V, prove that the linear spanof S is a sub space of V.
8. Let V be an inner product space. If u, v Î V, prove that | (u, v)| £ || u || || v ||
9. Prove that a semi lattice satisfying absorption laws is a lattice.
10. In any distributive lattice, prove that the set of all complemented elements forms a sub lattice
2
PART B (3 × 20 = 60) Answer any THREE questions.
Each question carries TWENTY marks.
11. (a) If H and K are finite subgroups of a group G, of orders O(H) and O(K) respectively, prove that
(b) Let N be a subgroup of a group G. Prove that N is a normal subgroup of G if and only if every left coset of N in G is a right coset of N in G.
12. (a) If P is a prime number and P divides the order of G, prove that G has an element of order P.
(b) If P is a prime number and Pa divides the order of G, prove that G has a subgroup of order Pa.
13. Prove that every integral domain can be imbedded in a field.
14. (a) If v1, v2 , ......, vn is a basis of V over F and if w1 , w2 , ....., wn in V are linearly independent over
F, prove that m £ n.
(b) Let A(W) denote the annihilator of W. Prove that A(A(w)) = W.
15. (a) Prove the following relations for any Boolean lattice : (a¢)¢ = a,
(a L b) ¢ = a¢ V b¢, (a V b)¢ = a¢ L b¢.
(b) Prove the following : ? Every finite Boolean algebra L is isomorphic to 2n for some n Î N ?
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