LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – NOVEMBER 2012

MT 4502/4500 – MODERN ALGEBRA

Date: 03-11-2012                       Dept. No.                                                       Max. : 100 Marks

Time: 1.00 – 4.00

                                                                 

                                                                  SECTION – A                                       (10 ´ 2 = 20)

Answer ALL the questions.

1. Define partially ordered set and give an example.
2. Show that identity element of a group is unique.
3. Let and . Prove that divides .
4. Show that any group of order up to 5 is abelian.
5. Define kernel of a homomorphism.
6. State fundamental theorem of homomorphism.
7. Define a ring and give an example.
8. If is a ring with unit element , then for all , show that .
9. State unique factorization theorem.
10. If is a commutative ring with unity, prove that every maximum ideal of  is a prime ideal.

 

PART – B ( 5 ´ 8 = 40)

Answer any FIVE questions

11. If is a group in which  for three consecutive integers  for all , show that  is abelian.
12. Show that a subgroup of a group  is a normal subgroup of  if and only if every left coset of  in  is a right coset of in .
13. `Prove that every group of prime order is cyclic.
14. State and prove Cayley’s theorem.
15. Show that any two finite cyclic groups of the same order are isomorphic.
16. Define a subring of a ring. Show that the intersection of two subrings of a ring  is a subring of .
17. Show that every finite integral domain is a field.
18. Show that every Euclidean ring is a principal ideal domain.

 

PART – C (2 ´ 20 = 40)

Answer any TWO questions

19. (a) State and prove the fundamental theorem of arithmetic.

(b)  Show that a nonempty subset  of a group  is a subgroup of  if and only if implies .

20. (a) State and prove the Lagrange’s theorem.

(b)   Let  be a commutative ring with unit element whose only ideals are  and  itself.  Show that  is a field.

21. (a) Determine which of the following are even permutations:

• (ii)

(b)   If  is a group, then show that , the set of automorphisms of , is also a group.

22. (a) Show that an ideal of the Euclidean ring  is a maximal ideal of  if and only if it is generated by a prime element of .

(b)   Show that , the set of all Gaussian integer, is a Euclidean ring.

 

 

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