Test: Group Theory - 6 - Mathematics MCQ

m = n

m ≠ n

m = n = 0

None of these

finite

infinite

zero

None of these

isomorphism

homomorphism

automorphism

None of these

zero

one

infinity

None of these

xa = ax

xa = e

ax = e

xa ≠ ax

f = ( 12) (1 3) (1 4) (2 5)

g = (1 2 3 4 5) (1 2 3) (4 5)

h = ( 1 2 3) (1 2 )

None of the above

 even permutation

odd permutation

neither even nor odd

None of these

2

3

5

None of these

H is isomorphic to K, since both are cyclic.

H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1

H is not isomorphic to K, since there is no homomorphism from H to K. 

H is not isomorphic to K, since K is cyclic where as H is not.

G is abelian

G is non cyclic

G is non abelian

centre of G has order 7

Every quotient group of a group is abelian and its converse is also true.

Every quotient group of a group is abelian but its converse is not true

Every quotient group of an abelian group is abelian and its converse is also true.

Every quotient group of an abelian group is abelian but the converse is not true.

H is a normal subgroup of G

H is not a normal subgroup of G

H is a subgroup of G

None of the above

(Z +)

(N, +)

(R, +)

None of these

singular

non-singular 

inverse

multiplicative

semigroup

subgroup

cyclic group 

abelian group

HK is a subgroup but H ∩ K may not be a subgroup.

H  ∩ K is a subgroup but HK may not be a subgroup.

Either (a) or (b).

None of the above.

1

2

3

4

N₁ ⊂ N₂

N₂ ⊂ N₁

N₁ = N₂

N₁,N₂ ⊂ N₁

1

2

5

10