Tamil Nadu Open University (TNOU) 2009-2nd Year B.Sc Mathematics " GROUPS AND RINGS " UG 478 BMS 21 - Question Paper
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UG-478 BMS-21B.Sc. DEGREE EXAMINATION -JANUARY 2009.
Second Year Mathematics GROUPS AND RINGS
Time : 3 hours Maximum marks : 75
PART A (5 x 5 = 25 marks)
Answer any FIVE questions.
1. Show that / : R > R defined by / (x) = 2x - 3 is a bijection and find its inverse. Compute f~l f and f of-1.
2. State and prove second principle of induction.
3. If A is the set of all even permutations in Sn,
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UG-478 BMS-21n
then prove that A is a group containing permutations.
4. Prove that a subgroup of a cyclic group is cyclic.
5. If the index of a subgroup H of a group G is two, then show that aH = Ha, for every a eG.
6. If G is any group and aeG, then show that (Z>a : G > G defined by <fta (x) = axa-1 is an automorphism of G.
7. Prove that any finite integral domain is a field.
8. Prove that any Euclidean domain R has an identity element.
PART B (5 x 10 = 50 marks)
Answer any FIVE questions.
9. If f\A>B and g:B>C are bijections, then show that (g /)_1 = /_1 g~x.
10. If A and B are two subgroups of a group G, then prove that AB is a subgroup of G if and only if AB = BA.
11. If G is a group and a,6eG, then show that
(a) order of a = order of a-1
(b) order of a = order of b~lab
(c) order of ab = order of ba.
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UG-478 BMS-2112. State and prove Lagranges theorem. Discuss about its converse.
13. If G is a cyclic group generated by a and f:G>G is a mapping such that / (xy) = f (x) / (y), then prove that f is an automorphism of G if and only if f (a) is a generator of G.
14. If R is a commutative ring with identity, then prove that every maximal ideal of R is prime ideal of R.
15. Prove that for any prime p, Zp is not an ordered integral domain.
16. Show that the ring of Gaussian integers is an Euclidean domain.
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