**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.**

- Define partially ordered set and give an example.
- Show that identity element of a group is unique.
- Let and . Prove that divides .
- Show that any group of order up to 5 is abelian.
- Define kernel of a homomorphism.
- State fundamental theorem of homomorphism.
- Define a ring and give an example.
- If is a ring with unit element , then for all , show that .
- State unique factorization theorem.
- 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**

- If is a group in which for three consecutive integers for all , show that is abelian.
- 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 .
- `Prove that every group of prime order is cyclic.
- State and prove Cayley’s theorem.
- Show that any two finite cyclic groups of the same order are isomorphic.
- Define a subring of a ring. Show that the intersection of two subrings of a ring is a subring of .
- Show that every finite integral domain is a field.
- Show that every Euclidean ring is a principal ideal domain.

__PART – C__ (2 **´ 20 = 40)**

**Answer any TWO questions**

- (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 .

- (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.

- (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.

- (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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