Test: Group Theory - 3 | 20 Questions MCQ Test

QUESTION: 1

If N is a set of natural numbers, then under binary operation a · b = a + b, (N, ·) is

A.
quasi-group
B.
semi-group
C.
monoid
D.
group

Solution:

QUESTION: 2

The number of generators in cyclic group of order 10 are

A.
3
B.
1
C.
2
D.
4

Solution:

QUESTION: 3

The set of all positive rational numbers forms an abelian group under the composition defined by

Solution:

QUESTION: 4

Set (1,2,3,4} is a finite abelian group of order... under multiplication modulo ... as composition.

A.
3,4
B.
4, 5
C.
1 , 2
D.
2 , 3

Solution:

QUESTION: 5

Let G be a group of order 7 and φ(x) = x⁴, x ∈ G. Then f is

A.
not one-one
B.
not onto
C.
not a homomorphism
D.
one-one, onto and a homomorphism

Solution:

A group of prime order must be cyclic and every cyclic group is abelian. Then we can show that φ: G → G s.t. φ(x) = xⁿ is an isomorphism if 0(G) and n and are co-prime.

QUESTION: 6

HK is a sub-group of G iff

A.
HK = KH
B.
HK ⊂ KH
C.
HK ⊃ KH
D.
HK ≠ KH

Solution:

QUESTION: 7

Check the correct statement.

A.
Every subgroup of a cyclic group is cyclic.
B.
If G is an infinite cyclic group, then G has exactly two generators and G is isomorphic to the additive group of integers.
C.
Every finite group of composite order possesses proper subgroups.
D.
All of the above.

Solution:

QUESTION: 8

If a, b ∈ G, a group, then b is conjugate to a, if there exist c ∈ G, such that

A.
b = c^-1ac
B.
a = cb
C.
b = ac^-1
D.
b = cc^-1a

Solution:

QUESTION: 9

If H₁ and H₂ are two subgroups of G, then following is also a subgroups of G

A.
H₁∩H₂
B.
H₁∪ H₂
C.
H₁H₂
D.
None of these

Solution:

QUESTION: 10

If (G, *) is a group and for all a, b ∈ G, b^-1* a^-1* b * a = e, then G is

A.
abelian group
B.
non group
C.
ring
D.
field

Solution:

QUESTION: 11

Number of elements of the cyclic group of order 6 can be used as generators of the group are

A.
= 3
B.
= 5
C.
= 2
D.
4

Solution:

Here, 6 = 2 x 3

QUESTION: 12

The multiplicative group {1, -1} is a subgroup of the multiplicative group

A.
{1, i, -i}
B.
{1,-1, i, -i}
C.
{1,0, -1, -i}
D.
{-1, i, -i}

Solution:

QUESTION: 13

A set G with a binary composition denoted multiplicative is a group, if

A.
the composition is associative
B.
for every pair o f elements a, b ∈ G, the equation ax = b and ya = b have solution in G
C.
Both (a) and (b)
D.
None of the above