Notesale: Turn your study into money

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Time Allowed: 35 Minutes
(OBJECTIVE PART)
1- a) Tick or Encircle the correct answer:
i)

ii)

a) H1 ∩ H2 = ф

b) H1 ∩ H2 = H1

c) H1 U H2 = H1

d) Either (b) or (c) is true
...
Then dim of W is
a) 1

iv)

Sign of
Supdt
...
Then H1 U H2 is a subgroup of G if and only if

a) 2Z/
iii)

Max
...

a) Integral Domain

b) Division Ring

c) Field

d) Cumulative Ring

b) Indicate True or False:

1x8

i)

(3 Z/ , +,
...


True / False

ii)

The generators of the cyclic group (Z/ , +) are 1 and -1
...
Then G has a subgroup of order r iff r is even
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True / False

v)

The ring Z/ 6 = {0, 1, 2, 3, 4, 5} with respect to addition and multiplication
module 6 contain no zero divisor
...


True / False

vii)

Every basis is maximal linearly independent set
...


2

True / False

c) Fill in the blanks meaningfully:

1x4

i)

A group homomorphism ф is injective if and only if ____________________________________

ii)

A commutative division ring is ____________________________________________________

iii)

Centre of an abelian group is ______________________________________________________

iv)

If R is a commutative ring such that a ≠ 0 and ∃ b ∈ R s
...

(Continued Overleaf)

2- Give short answers the following questions:
2x8
i)
Define Rank and Nullity of a Vectorspace
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Write all the subgroups of G
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_______________________________________________________________________________________
_______________________________________________________________________________________
_______________________________________________________________________________________
iv)

If H and K are two sylow p-subgroups of a group G, describe the relation between H and K
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_______________________________________________________________________________________
_______________________________________________________________________________________
_______________________________________________________________________________________
vii)

4

4

If T : IR → IR is a linear Transformation and dim (R(T)) = 4
...


_______________________________________________________________________________________
_______________________________________________________________________________________
_______________________________________________________________________________________
***M
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Sc-I(11/A) (MTH-II) (N)***

Total Marks: 68 + 32 = 100
Pass Marks: 40%

SUBJECTIVE PART

SECTION-A
~ HK /
3- a) Let G be a group, H a subgroup and K a normal subgroup of G then show that H / H∩K =
K 9
b) Prove that a group of prime order is abelian
...
4,1,3
4- a) Define a Permutation and a Transposition
...
Specify which one is normal and
which one is not and why?
2,3,3
b) Let H and K be subgroups of a group G
...

9
5- a) If G is a group of order n divisible by a prime p
...

b) Prove that Centre of a finite p-group is non-trivial
...

b) Let R, R′ be rings and ф : R→ R′ be a ring hamomorphism
...


7- a) Suppose T : V → W is a linear transformation from a finite dimensional vector space V into a
vector space W
...

b) If V and W are of dimension m and n respectively over F
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8- a) Define Similar Matrices
...

1
4
b) Check the matrix
A=
3
2
-1
for diagonalizability
...

***M
...
Sc-I(11/A) (MTH-II) (N)***

8

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8

8

9



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