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.
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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