LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS

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 a^{n} = 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 a^{5} = e, b^{4} = e, ab = ba^{3}, prove that
(i) a^{2}b = ba and (ii) ab^{3} = b^{3}a^{2}.
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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