# Loyola College B.Sc. Mathematics April 2009 Modern Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

 ZA 16

FOURTH SEMESTER – April 2009

# MT 4502 / 4500 – MODERN ALGEBRA

Date & Time: 24/04/2009 / 9:00 – 12:00  Dept. No.                                                   Max. : 100 Marks

PART – A

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

1.)     Define an equivalent relation on set A.

2.)     Define a partially ordered set.

3.)     Define a cyclic group.

4.)     If G is a group of order n and a Î G, show that an = e.

5.)     Define an automorphism of a group.

6.)     Define the alternating group of degree n.

7.)     Define a ring.

8.)     Define a field.

9.)     Define an ordered integral domain.

10.)   Define a maximal ideal of a ring.

PART – B

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

11.)   If G is a group, prove that

a.) the identity element of G is unique

b.) every a Î G has a unique inverse in G.

12.)   Show that the union of two subgroups of G is a subgroup of G if and only if one is contained in the other.

13.)   Prove that every subgroup of a cyclic group is cyclic.

14.)   If a and b are elements of a group and a5 = e, b4 = e, ab = ba3, prove that

(i)  a2b = ba        and               (ii) ab3 = b3a2.

15.)   Show that a subgroup N of a group G 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.

16.)   If f is a homomorphism of a ring R into a ring R, then prove that the kernel of f is an ideal
of R.

17.)   Prove that the intersection of two subrings of a ring is a subring.

18.)   Prove that an element a in a Euclidean ring R is a unit if and only if d(a)= d(1)

PART – C

Answer any two questions:                                                                                            (2 x 20 = 40)

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

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

(b) Let H be a subgroup of index 2 in a group G.  Prove that H is a normal subgroup.  (15+ 5)

21.) (a) State and prove the fundamental theorem of homomorphism on groups

(b) Define an integral domain and a division ring                                                            (14+ 6)

22.)   State and prove unique factorization theorem.

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