LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc.** DEGREE EXAMINATION – **MATHEMATICS**

FOURTH SEMESTER – **APRIL 2012**

# MT 4502 – MODERN ALGEBRA

Date : 21-04-2012 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

__PART – A__

**ANSWER ALL THE QUESTIONS: (10 x 2 = 20 marks)**

- Define an equivalence relation.
- Show that if every element of the group G is its own inverse then G is abelian.
- Show that every cyclic group is abelian.
- Define a normal subgroup of a group
- Define the kernel of a homomorphism of a group.
- Express (1,3,5) (5,4,3,2) (5,6,7,8) as a product of disjoint cycles.
- If A is an ideal of a ring R with unity and 1A show that A=R.
- If F is a field show that its only ideals are and F itself.
- Define a maximal ideal.
- Show that every field is a principal ideal domain.

__PART – B__

**ANSWER ANY FIVE QUESTIONS: (5 x 8 = 40 marks)**

- Show that a non empty subset H of a group G is a subgroup of G if and only if a,bH implies that ab
^{-1} - Show that every subgroup of a cyclic group is cyclic.
- State and prove Langrange’s theorem.
- 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.
- State and prove Cayley’s theorem.
- Show that every finite integral domain is a field.
- Show that every Euclidean ring is a principal ideal domain.
- Let R be a Euclidean ring. Show that any two elements a and b in R have a greatest common divisor d which can be expressed in the form d=λa+mb for l, m in R.

__PART –C__

**ANSWER ANY TWO QUESTIONS: (2 X 20 = 40 Marks)**

- (i) If H and K are finite subgroup of a group G, show that

(ii) Show that a group G cannot be the union of two proper subgroups. (12+8)

- (i) If G is a group and N is a normal subgroup of G, show that G/N, the set of all distinct left

cosets of N in G is also a group.

(ii) If H is the only subgroup of order o (H) in the group G, show that H is normal in G. (12+8)

- (i) If H and N are subgroups of a group G and suppose that N is normal in G, show that is

isomorphic to .

(ii) If R is a commutative ring with unit element whose only ideals are (O) and R itself, show that

R is a field. (10+10)

- (i) Show that an ideal of the Euclidean ring R is a maximal ideal if and only if it is generated by a

prime element of R.

(ii) Show that the characteristic of an integral is either zero or a prime number. (14+6)

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