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B.Sc-MATHEMATICS 2nd Year ABSTRACT ALGEBRA AND REAL ANALYSIS(Kakatiya University (KU), Warangal, Andhra Pradesh-2010)

KAKATIYA UNIVERSITY
B.A./B.Sc II YEAR MATHEMATICS
(2009-2010)
paper-II
ABSTRACT ALGEBRA AND REAL ANALYSIS
QUESTION BANK FOR PRACTICAL EXAMINATION

UNIT-I

1. a) Show that the set S = ( {1, 3, 5, 7}, X8) forms a group.

b) Show that every group G with identity e and such that x * x = e for all

x I G is abelian.

2. Find the order of the cyclic subgroup of U6 generated by cos 2p/3 + i sin 2p/3.

3. Find the order of the cyclic subgroup of the multiplicative group G of invertible

4 x 4 matrices generated by

4. Compute the subgroups <1>, <2>, <3>, <4> and <5> of the group Z6.

5. a) Find the number of generators of a cyclic group having the order 60.

b) Find the number of automorphisms of the group Z6.

6. Find the number of elements in the cyclic subgroup of the group C* of nonzero complex numbers generated by 1 + i.

7. Find all subgroups of the group Z12, and draw the subgroup diagram for the subgroups of Z12.

8.

9. Find all orders of subgroups of the group Z12. Also, find the elements in each subgroup.

10. If s = , Z =

then find i) s -1 Z s ii) | < s > |

11 a) Find all orbits of

b) Express the following permutation as a product of disjoint cycles, and then as a product of transpositions

12. a) Find all orbits of the permutation s, where s : Z ® Z is given by
s (n) = n + 1

b) What is the order of Z = (1, 4) (3, 5, 7, 8) ?

13. Find all cosets of the subgroup 4Z of Z. Also, find (Z : 4Z).

14. Find all cosets of the subgroup < 4 > of Z12. What is the index of < 4 > in Z12.

15. Find the index of < 3 > in the group Z24.

16. Let s = (1, 2, 5, 4) (2, 3) in S5. Find the index of < s > in S5.

17. a) Let f : R* ® R* under multiplication be given by f (x) = | x |. Show that f is a homomorphism.

b) Let ker (f) and f (25) for f : Z ® Z7 such that f(1) = 4.

18. a) Let f : R ® R*, where R is additive group and R* is multiplicative group, be given by f (x) = 2x. Examine whether f is a homomorphism or not.

b) Find ker (f) and f (18) for f : Z ® Z10 such that f(1) = 6.

UNIT-II

19. a). Find all units in the ring Z x Z.

b). Find the solutions of the equation x2 + x - 6 = 0 in the ring Z14 by factoring the quadratic polynomial.

20. a). Find all units in the ring Z5.

b). Find all solutions of the equation x3 - 2x2 - 3x = 0 in Z12.

21. a). Find all units in the ring Z x Q x Z.

b). Solve the equation 3x = 2 in the field Z7

22. a). Find all units in the ring Z4.

b). Find the Characteristic of the ring Z3 x Z4.

23. a). Find all units in the matrix ring M2(Z2).

b). Find the Characteristic of the ring Z x Z.

24. a). Find all solutions of the equation x2+2x+2 = 0 in Z6.

b). Find the characteristic of the ring Z3 x 3Z.

25. a). Find the characteristic of the ring 2Z.

b). Let R be a commutative ring with unity of characteristic 4. Compute and simplify (a + b)4 for a, b ? R.

26. a). Find the characteristic of the ring Z6 x Z15.

b). Let R be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for a, b ? R.

27. a). Find the characteristic of the ring Z3 x Z3.

b). Find all prime ideals and all maximal ideals of Z2 x Z2.

28. a). Find all solutions of x2 - 2x + 4 = 0 in Z6.

b). Find all prime ideals and all maximal ideals of Z6.

29. a). Find all prime ideals and all maximal ideals of Z12.

b). Find a prime ideal of Z x Z that is not maximal.

30. Show that for a field F , the set S of all matrices of the form for a, b ? F

is a right ideal but not a left ideal of M2(F). Is 'S' a sub ring of F?

31. Let A and B be ideals of a commutative ring R. The quotient A : B of A by B is

defined by A: B { r ? R / r b ? A for all b ? B}.

Show that A: B is an ideal of R.

32. Let R and R' be two rings and let f : R® R' be a ring homomorphism such that f(R) ? {0' }. Show that if R has unity 1 and R' has no zero devisors then f(1) is unity for R'.

33. If R is a ring with unity and N is an ideal of R such that N? R then find the unity of R/N.

34. Let R be commutative ring and let aIR. Show that Ia = {xIR/ax = 0}is an ideal of R.

35. Let R be a ring with contains at least two elements. Suppose for each non zero a IR, there exists a unique b IR such that aba =a.

(a) Show that R has no devisors of zero.

(b) Show that bab = b

36. Let A and B are ideals of a ring R, the sum A+B of A and B is defined by A+B={a+b/aIA, b I B}.

(a) Show that A+B is an ideal (b) Show that A K A+B and B K A+B.

UNIT-III

37. a) Use the definition of the limit of a sequence to establish

b) If 0 < a < b, determine

38. Determine the limits of

39. Let x1 = 8 and xn+1 = ½ xn + 2 for n I N. Show that (xn) is bounded and monotone. Find the limit.

40. Let x1 > 1 and xn+1 = 2 - 1/xn for n I N. Show that (xn) is bounded and monotone. Find the limit.

41. Let x1 = 1 and xn+1 = for n I N. Show that (xn) converges and find the limit.

42. Let for each n I N. Prove that (xn) is increasing and bounded, and hence converges.

43. Establish the convergence and find the limits of the sequences

i) ii)

44. a) Establish the convergence and find the limit of

b) Determine the limit of

45. a) Establish the convergence and find the limit of

b) Determine the limit of

46. a) Show that the sequence is a Cauchy sequence.

b) Show that is not a Cauchy sequence.

47. If x1 < x2 are arbitrary real numbers and for n > 2, show that (xn) is convergent. What is its limit?

48. a) Show that .

b) Show that the series is convergent.

49. a) Show that

b) Show that the series is convergent.

50. Use the Cauchy condensation test to

a) Establish the divergence of

b) Establish the convergence of

51. a) Establish the convergence of by comparison test.

b) Establish the convergence of by ratio test.

52 Find the value of k such that f(x) is continuous at x = 2.

a) Let K > 0 and let f : R ® R satisfy the condition | f(x) - f(y) } < K | x - y |, for all x, y I R. Show that f is continuous at every point c I R.

b) Suppose f : R ® R is continuous on R and f (r) = 0 for every rational number r. Prove that f(x) = 0 for all x I R.

53 A function f : R ® R is said to be additive if f(x+y) = f(x) + f(y) for all x, y in R. Prove that if f is continuous at some x0, then it is continuous at every point of R.

54 Let I = [a, b] and let f : I ® R and g : I ® R be continuous on I. Show that the set E={ x I I : f(x) = g(x) } has the property that if (xn) K E and xn ® x0, then x0 I E.

UNIT-IV

55 a) Show that f(x) = x1/3, x I R, is not differentiable at x = 0.

b) Let f : R ® R be defined by f(x) = x2 for x rational

0 for x irrational.

Show that f is differentiable at x = 0, and find f ' (o).

56 a) Find the points of relative extrema, the intervals on which the function

f(x) = x + 1/x, (x ¹ 0) is increasing or decreasing.

b) Use the mean value theorem to show that | sin x - sin y | < | x - y |, x, y I R.

57 a) Let a1, a2 …… an be real numbers and let f be defined on R by for x I R. Find the unique point of relative minimum for f.

b) Use the mean value theorem to show that

58 a) Let f(x)= x2 sin 1/x for x ¹ 0, let f(o) = 0, and let g(x) = sin x for x IR. Show that = 0, but does not exist.

b) Find

59 a) Find

b) Find

60 a) Find

b) Find

61 Show that if x > 0, then .

62 Use Taylor's theorem with n = 2 to obtain more accurate approximations for Ö1.2 and Ö2.

63 a) If f(x) = ex, show that the remainder term in Taylor's theorem converge to zero as n ® µ, for each fixed x0 and x.

b) Determine whether or not x = 0 is a point of relative extremum of .

64 a) If f(x) = sin x, show that the remainder term in Taylor's theorem converges to zero as n ® µ for each fixed x0 and x.

b) Determine whether or not x = 0 is a point of relative extremum of .

65 Let f(x ) = 2 if 0 < x < 1

1 if 1 < x < 2.

Show that f I R [0, 2] and evaluate its integral.

66 a) If f I R [a, b] and | f (x) | < M for all x I [a, b], show that

b) Let g(x) = 0 if x I [0, 1] is rational.

1/x if x I [0, 1] is irrational.

Explain why g L R [0, 1]. However, show that there exists a sequence (Pn) of tagged partitions of [a, b] such that || Pn || ® 0 and exists.

67 a) Consider the function h defined by h(x) = x + 1 for x I [0, 1] rational, and h(x) = 0 for x I [0, 1] irrational. Show that h is not Riemann integrable on [0, 1].

b) Suppose that f is continuous on [a, b], that f(x) > 0 for all x I [a, b] and that . Prove that f(x) = 0 for all x I [a, b].

68 Suppose that a > 0 and that f I R [-a, a].

a) If f is even for all x I [0, a], show that

b) If f is odd, for all x I [0, a], show that .

69 a) If n I N and for x I [a, b], show that the fundamental theorem implies that . What is the set E here ?

b) If g(x) = x for | x | > 1

-x for | x | < 1

and if G(x) = ½ |x2 - 1 |, show that .

70 a) Let B(x) =

Show that .

b) Let f I R [a, b] and define .

Then evaluate in terms of F.

71 a) Find F' (x) when F is defined on [0, 1] by : .

b) If f : R ® R is continuous and c > 0, define g : R ® R by

Show that g is differentiable on R and find g' (x).

72 a) Find F' (x) when F is defined on [0, 1] by :

b) If f : [0, 1] ® R is continuous and for all x I [0, 1], show that f(x) = 0 for all x I [0, 1].