LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ  13

FOURTH SEMESTER – APRIL 2008

MT 4502 / 4500 – MODERN ALGEBRA

 

 

 

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

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

1. Define an equivalence relation on a set.
2. Define a binary operation on a set.
3. Define a cyclic group.
4. Define a quotient group of a group.
5. Define an isomorphism.
6. Define a permutation group.
7. Define a division ring.
8. Define a field.
9. Define an integral domain.
10. What is a Gaussian integer?

 

PART – B

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

1. If G is a group, then prove that

• for every
• for all

1. Prove that anon – empty subset H of a group G is a subgroup of G if and only if

(i)

(ii)

1. If H is a subgroup of a group G, then prove that any two left Cosets of H in G either are identical or have no element in common.
2. If H is a subgroup of index 2 in a group G, prove that H is a normal subgroup.
3. If  is a homomorphism of a group G into a group , prove that

(i) , the identity element of G¹

(ii) for all

1. Show that the additive group G of integers is isomorphic to the multiplicative group
2. Prove that the intersection of two subrings of a ring R is a subring of R.
3. Find all the units in Z(i).

 

PART – C

Answer any TWO   questions.                                                                      (2 x 20 = 40)

1. State and prove the Fundamental theorem of arithmetic.
2. a) State and prove Lagrange’s theorem.

1. b) Show that every subgroup of an abelian group is normal. (14+6)

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

1. b) Define an endomorphism an epimorphism and an automorphism.

1. State and prove unique factorization theorem.

 

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       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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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.

1. 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.
2. Define a partially ordered set and give an example.
3. Show that the intersection of two normal subgroups is again a normal subgroup.
4. Give an example of an abelian group which is not cyclic.
5. 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.

6. 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.

7. 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.

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

SECTION-B                                                             (5X8=40)

Answer any FIVE questions.

11. 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.
12. 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.

13. 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.

14. Prove that any group is isomorphic to a group of permutations.
15. 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.

16. Show that Qis a field under the usual addition and multiplication.
17. 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.

18. Show that every finite integral domain is a field.

SECTION-C                                                       (2X20=40)

Answer Any Two

19. a) If H and K are finite subgroups of a group G then prove that  o(HK)= o(H)o( K)/o(H
20. b) Prove that every subgroup of a cyclic group is cyclic.                            (12+8)
21. 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.

1. b) State and prove Euler’s theorem and Fermat’s theorem.                                                         (10+10)
2. a) State and prove Fundamental homomorphism theorem for groups.
3. 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)

22. a) State and prove unique factorization theorem.
23. 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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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)

 

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

 

PART – B

 

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

 

11. 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
12. Show that every subgroup of a cyclic group is cyclic.
13. State and prove Langrange’s theorem.
14. 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.
15. State and prove Cayley’s theorem.
16. Show that every finite integral domain is a field.
17. Show that every Euclidean ring is a principal ideal domain.
18. 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)

 

19. (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)

 

 

20. (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)

 

21. (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)

 

22. (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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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.

1. Define partially ordered set and give an example.
2. Show that identity element of a group is unique.
3. Let and . Prove that divides .
4. Show that any group of order up to 5 is abelian.
5. Define kernel of a homomorphism.
6. State fundamental theorem of homomorphism.
7. Define a ring and give an example.
8. If is a ring with unit element , then for all , show that .
9. State unique factorization theorem.
10. 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

11. If is a group in which  for three consecutive integers  for all , show that  is abelian.
12. 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 .
13. `Prove that every group of prime order is cyclic.
14. State and prove Cayley’s theorem.
15. Show that any two finite cyclic groups of the same order are isomorphic.
16. Define a subring of a ring. Show that the intersection of two subrings of a ring  is a subring of .
17. Show that every finite integral domain is a field.
18. Show that every Euclidean ring is a principal ideal domain.

 

PART – C (2 ´ 20 = 40)

Answer any TWO questions

19. (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 .

20. (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.

21. (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.

22. (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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1502 – ALGEBRA AND CALCULUS – I

 

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

ANSWER ALL QUESTIONS:                                                                (10×2=20)

 

1. Find the n^th derivative of
2. Show that, in the curve, the polar sub tangent varies as the square of the

radius vector and the polar subnormal is a constant.

3. Write the conditions for maxima and minima of two variables.
4. What is the radius of curvature of the curve at the point (1, 1)?
5. Find the co-ordinates of the centre of curvature of the curve at the point

(2, 1).

6. Form a rational cubic equation which shall have the roots 1,
7. If are the roots of the biquadratic equation

find

8. State Newton’s theorem on the sum of the powers of the roots.
9. State Descartes’ rule of signs for negative roots.
10. Determine if Cardon’s method can be applied to solve the equation

 

PART – B                               

 

ANSWER ANY FIVE QUESTIONS:                                                      (5×8=40)

11. a) Find the angle at which the radius vector cuts the curve
12. b) Find the slope of the tangent with the initial line for the cardioid

at                                                                                            (4 + 4)

12. Discuss the maxima and minima of the function

 

13. Prove that the (p-r) equation of the cardioid is and hence

prove that its radius of curvature is

14. Show that the evolute of the cycloid ; is another

cycloid.

15. Solve the equation
16. Show that the sum of the eleventh powers of the roots of is zero.
17. a) If are the roots of the equation find the value of

 

1. b) Determine completely the nature of the roots of the equation

(5 + 3)

18. If be a real root of the cubic equation of which the coefficients

are real, show that the other two roots of the equation are real, if

 

PART – C                               

 

ANSWER ANY TWO QUESTIONS:                                                      (2 x 20 = 40)

 

19. a) Find the n^th differential coefficient of .
20. b) If, prove that               (10 +10)

 

20. A tent having the form of a cylinder surmounted by a cone is to contain a given

volume. If the canvass required is minimum, show that the altitude of the cone is

twice that of the cylinder.

 

21. a) Find the asymptotes of

 

1. b) Show that the roots of the equation are in Arithmetical

progression if  Show that the above condition is satisfied by the

equation and hence solve it.                                              (10 + 10)

 

22. Determine the root of the equation which lies between 1 and 2

correct to three places of decimals by Horner’s method.

 

 

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