Download OU (Osmania University) B.Sc Computer Science 4th Sem Algebra Important Question Bank For 2021 Exam

Subject Title: Algebra

Prepared by: S Shravani

Year: II

Semester: IV

Updated on: 23.03.

Unit - I

1.

A group G, identity element is unique.

2.

In a group G, inverse element is unique.

3.

Prove that the set Z of all integers form an abelian group w.r.t the operations defined by a*b =

a+b+2, for all a,b Z

4.

Cancellation laws holds in a group. (let G be a group. Then for a,b, c G, ab=ac b = c and

ba=ca b = c.

5.

In a group G for a,b,x,y G the equation ax = b and ya = b have unique solutions.

6.

If every element of a group (G,.) is its own inverse , show that (G,.) is abelian group.

7.

The order of every element of a finite group is finite and less than or equal to the order of a

group.

8.

In a group G ifa G, then |a|= |a-1|.

9.

If a is an element of group G such that |a| = n, then an=eiff n/m.

10. If a is an element of group G such that |a| = 7, then show that a is the cube of some element of

G.

11. A non-empty complex H of a group G is subgroup pf G iff (i) a H, b H ab H

a. (ii) a H,a-1 H.

12. A non-empty complex H of a group G is subgroup pf G iffa H, b H ab-1 H

13. The necessary and sufficient condition for a finite complex H of a group G to be a subgroup of

G is a H, b

H ab H.

14. If H and K are two subgroups of a group G, then HK is subgroup of G iff HK = KH

15. If H1 and H2 are two subgroups of a group G, then H1 H2is also a subgroup of G.

16. The union of two subgroups of a group G is a subgroup iff one is contained in other.

17. Every cyclic abelian group is an abelian group. Converse is not true.

18. Every subgroup of cyclic group is cylic.

19. If a cyclic group G is generated by an element of order n, the an is generator of G iff (m,n) = 1.

20. The order of a cyclic group is equal to its generator.

21. Show that the group ( G= {1,2,3,4,5,6} ,?7 ) is cyclic . also write down all its generators.

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22. How many subgroups does Z20 have? List them.

23. If G is an infinite cyclic group , the G has exactly two generators which are inverse of each

other.

24. Write down the following products as disjoint cycles.

(i)

(1 3 2) (5 6 7) (2 6 1) ( 4 5)

(ii) (ii) (1 3 6) (1 3 5 7) (6 7) (1 2 3 4).

25. Definations:

i.

Group

ii.

Subgroup

iii.

Addition modulo

iv.

Multiplication modulo

v.

Cyclic group

vi.

Permutation group

vii.

Order of a group

viii.

Order of an element

Unit -III

26. If R is a Boolean ring then (i) a+a=0 for all a R (ii) a+b = 0 a=b and

**(**iii) R is commutative

under multiplication.

27. Find all units of Z14

28. The intersection of two subring of a ring R is a subring of R, A ring has no zero divisors iff the

cancellation laws holds in R, A field has no zero-divisors.

29. List all zero-divisors in 20 Can you see a relationship between the zero-divisors of 20 and the

units of 20 ?, A field is an integral domain.

30. Every finite integral dimain is a field.

The characteristic of an integral domain is either a prime or zero.

31. A field has no proper nin-trivial ideals.

A commutative ring R with unity element is a field if R have no proper ideals.

The union of two ideals of a Ring R is a ideal iff one is contained in other.

Every ideal of z is a principal ideal.

32. An ideal U of a commutative ring R with unity is maximal iff the quotient eing R/U is a field.

Find all maximal ideals in a. Z8. b. Z10. c. Z12. d. Zn.

33.

1. Definations:

Ring, Boolean ring, Sub ring

Integral domain

Field

Zero divisor

Characteristic of a ring

Idempotent and nilopotent

Ideal

Principal ideal

Maximal ideal

Factor ring

Prime ideals

Unit IV

34.

1. Test by divisibility by 9.

2. Prove that a ring homomorphism carries an idempotent to an idempotent.

3. The homomorphic image of a ring is a ring.

35. Let R, R

*be two rings and R R*

*be a homomorphism. Foe every ideal*

*in a ring R*

*,*

-1 (

*) is an ideal in R.*

36.

Let f(x) = 4x3 + 2x2 + x + 3 and g(x) = 3x 4 + 3x 3 + 3x 2 + x + 4, where f(x), g(x) Z5[x].

Compute f(x) + g(x) and f(x).g(x).

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37. If is a homomorphism of a ring R into a ring R

*the ker is an ideal of R.*

If is a homomorphism of a ring R into a ring R

*then R*

*then is an into isomophism iff*

Every quotient ring of a ring is a homomorphic image of a ring.

Fundamental theorem of homomorphism

38. The division algorithm.

Factor theorem.

Determine all ring homomorphisms from Z to Z.

39. Definitions:

i. Homomorphism ring

ii. Isomorphism ring

iii. Homomorphic image of ring

iv. Monomorphism ring

v. Automorphism ring

vi. Kernel of a homomorphism of ring

vii. Polynomial ring

viii. Degree of polynomial

Unit II

40. Any infinite cylic group is isomorphic to integers Z.

Cayley's theorem.

Find the regular permutations group isomorphic to the multiplicative group

{1, -1, , - }.

41. Let (G , .) , (G

*, .) be two groups. let be a homomorphism from G into G*

*then (i) (e) = e*

where e is the identity in G and e

*is identity in G. (ii) (a-1)= { (a)}-1.*

42. Every homomorphic image of an abelian group is abelian.

commute.

43. let be a isomorphism from G onto G

*then G = (a)iff G*

*= ( (a)) .*

44. let be a isomorphism from G onto G

*then for any elements a and b in G, a and b commute iff*

(a) and (b)

45. show that the mapping G G such that (a) = a-1 for all a G, is an automorphismof a

group G iff G is abelian.

The set of all automorphism of a group G forms a group w.r.t composition of mapping.

46. The set of all inner automorphism of a group G forms a group w.r.t composition of mapping.

H is any subgroup of a group (G , .) and h G then h H, iff hH=H=Hh.

If a and b are any two elements of group G and H is subgroup of group G then

47. Any two left (right) cosets of a subgroup are either disjoint or identical.

If H is a subgroup of a group G fora,b G the relation a b(modH) is an equivalence relation.

If H is a subgroup of a group G then there is one-one correspondence between the set of all

distinct left cosets of H in G and the set of all distinct right cosets of H in G.

48. Lagrange's theorem.

If G is a finite group and a G , then |a|/|G|.

If p is a prime number then every group of order p is cyclic group i.e a group of prime order is

cyclic.

Orbit ? Stabilizer theorem.

49. A subgroup of H of a group G is normal , if xH -1 = H for all x G.

A subgroup of H of a group G is a normal subgroup of G iff each left coset of H in G is a right

coset of H in G.

The set n of all even permutations on n symbols is a normal subgroup of the permutation

group Sn on the n symbols.

50. A subgroup of H of a group G is a normal subgroup of G iff product of two left(right) cosets of

H in G is again left(right) coset of H in G.

Every subgroup of an abelian group is normal.

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If G is a group and H is a subgroup of index 2 in G, then H is normal subgroup of G.

The union of two normal subgroups of a group G is a normal subgroup.

H is normal subgroup of G. the set G of all cosets of H in G w.r.t coset multiplication is a group

51.

1. Definitions:

i.

Homomorphism Group

ii.

Isomorphism Group

iii.

Homomorphic image of a Group

iv.

Monomorphism Group

v.

Automorphism Group

vi.

Coset

vii.

Congruence modulo

viii.

Index

ix.

Normal subgroup

x.

Factor group

xi.

Kernel of a homomorphism

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This post was last modified on 23 January 2021