Tamil Nadu Open University (TNOU) 2008-2nd Year B.Sc Mathematics "Moderan Algebra" UG 316 - Question Paper
UG-316 BMS-04
B.Sc. DEGREE EXAMINATION -JUNE 2008.
(AY 2005-2006, CY 2006 batches only)
Second Year
MODERN ALGEBRA
Time : 3 hours Maximum marks : 75
SECTION A (5 x 5 = 25 marks)
Answer any FIVE questions.
1. Show that the union of two equivalence relations need not be an equivalence relation.
2. Express the following permutation as a product of disjoint cycles :
r 1 2 3 4 5 6 7 [2 1 3 5 6 7 4J.
Prove that every cyclic group is abelian.
3.
4. Show that the centre H of a group G is a normal subgroup of G.
5. Prove that the intersection of two normal subgroups of a group G is a normal subgroup of G .
6. Show that any homomorphic image of a cyclic group is cyclic.
7. Show that any field F is an integral domain.
8. Prove that the characteristic of an integral domain D is either O or a prime number.
SECTION B (5 x 10 = 50 marks)
Answer any FIVE questions.
9. Prove that a non-empty subset H of a group G is a subgroup of G if and only if a, b e H ab 1 e H .
10. Prove that a subgroup of a cyclic group is cyclic.
11. State and prove Lagranges theorem on finite group.
12. If a group G has exactly one subgroup H of given order then show that H is a normal subgroup of G .
13. Show that isomorphism is an equivalence relation among groups.
14. State and prove the fundamental theorem of homomorphism.
15. Prove that the set F of all real numbers of the form a + b42 where a, b e Q is a field under the usual addition and multiplication of real numbers.
16. Prove that any integral domain D can be embedded in a field F and also every element of F can be expressed as a quotient of two elements of D .
3 UG-316
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