LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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

FOURTH SEMESTER – APRIL 2011

# MT 4502 – MODERN ALGEBRA

Date : 07-04-2011 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

SECTION-A (10X2=20) Answer ALL the questions.

- Let R be the set of all numbers. Define * by x*y=xy+1 for all x,y in R. Show that is commutative but not associative.
- Define a partially ordered set and give an example.
- Show that the intersection of two normal subgroups is again a normal subgroup.
- Give an example of an abelian group which is not cyclic.
- Let G be the group of non-zero real numbers under multiplication. and f:GG be defined

by f(x)=x for all xG. Is this map a homomorphism of G into G? Justify.

- If f is a homomorphism of a group G into a group G’ then prove that kernel of f is a

normal subgroup of G.

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

8 Let Z be the ring of integers. Give all the maximal ideals of Z.

- Show that every field is a principal ideal domain.
- Find all the units in Z[i]={x +iy/x,y Z}

SECTION-B (5X8=40)

Answer any FIVE questions.

- Prove that a non-empty subset H of a group G is a subgroup of G if and only if HH=H and H=H
^{-1}. - Let H be a subgroup of a group G. Then prove that any two left coset in G are either identical or have

no element in common.

- Show that a subgroup N of a group G is a normal subgroup of G iff every left coset of N in G is a

right coset of N in G.

- Prove that any group is isomorphic to a group of permutations.
- Prove that an ideal of the Euclidean ring R is a maximal ideal of R if and only if it is generated by a

prime element of R.

- Show that Qis a field under the usual addition and multiplication.
- Let R be an Euclidean ring. Then prove that any two elements a and b in R have a greatest common

divisor d which can be expressed by a + b.

- Show that every finite integral domain is a field.

SECTION-C (2X20=40)

Answer Any Two

- a) If H and K are finite subgroups of a group G then prove that o(HK)= o(H)o(
*K)/o(H* - b) Prove that every subgroup of a cyclic group is cyclic. (12+8)
- a) Prove that there is a one-one correspondence between any two left cosets of a subgroup

H in G and thereby prove the Lagrange’s theorem.

- b) State and prove Euler’s theorem and Fermat’s theorem. (10+10)
- a) State and prove Fundamental homomorphism theorem for groups.
- b) Let R be a commutative ring with unit element whose only ideals are (0) and R itself.

Prove that R is a field. (12+8)

- a) State and prove unique factorization theorem.
- b) Let R be the ring of all real valued functions on the closed interval [0,1].

Let M={f R/ f(1/2)=0}. Show that M is a maximal ideal of R. (10+10)

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