# Loyola College B.Sc. Mathematics April 2008 Modern Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

# XZ  13

FOURTH SEMESTER – APRIL 2008

# MT 4502 / 4500 – MODERN ALGEBRA

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions.:                                                                               (10 x 2 = 20)

1. Define an equivalence relation on a set.
2. Define a binary operation on a set.
3. Define a cyclic group.
4. Define a quotient group of a group.
5. Define an isomorphism.
6. Define a permutation group.
7. Define a division ring.
8. Define a field.
9. Define an integral domain.
10. What is a Gaussian integer?

PART – B

Answer any FIVE  questions.                                                                      (5 x 8 = 40)

1. If G is a group, then prove that
• for every
• for all
1. Prove that anon – empty subset H of a group G is a subgroup of G if and only if

(i)

(ii)

1. If H is a subgroup of a group G, then prove that any two left Cosets of H in G either are identical or have no element in common.
2. If H is a subgroup of index 2 in a group G, prove that H is a normal subgroup.
3. If  is a homomorphism of a group G into a group , prove that

(i) , the identity element of G1

(ii) for all

1. Show that the additive group G of integers is isomorphic to the multiplicative group
2. Prove that the intersection of two subrings of a ring R is a subring of R.
3. Find all the units in Z(i).

PART – C

Answer any TWO   questions.                                                                      (2 x 20 = 40)

1. State and prove the Fundamental theorem of arithmetic.
2. a) State and prove Lagrange’s theorem.
1. b) Show that every subgroup of an abelian group is normal. (14+6)
1. a) State and prove the fundamental theorem of homomorphism on groups.
1. b) Define an endomorphism an epimorphism and an automorphism.
1. State and prove unique factorization theorem.

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