Test: Group Theory - 3

# Test: Group Theory - 3

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## 20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Test: Group Theory - 3

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Test: Group Theory - 3 - Question 1

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

Detailed Solution for Test: Group Theory - 3 - Question 1 Test: Group Theory - 3 - Question 2

### The number of generators in cyclic group of order 10 are

Test: Group Theory - 3 - Question 3

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

Test: Group Theory - 3 - Question 4

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

Test: Group Theory - 3 - Question 5

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

Detailed Solution for Test: Group Theory - 3 - Question 5

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.

Test: Group Theory - 3 - Question 6

HK is a sub-group of G iff

Test: Group Theory - 3 - Question 7

Check the correct statement.

Test: Group Theory - 3 - Question 8

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

Test: Group Theory - 3 - Question 9

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

Test: Group Theory - 3 - Question 10

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

Test: Group Theory - 3 - Question 11

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

Detailed Solution for Test: Group Theory - 3 - Question 11

Here, 6 = 2 x 3 Test: Group Theory - 3 - Question 12

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

Test: Group Theory - 3 - Question 13

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

Test: Group Theory - 3 - Question 14

In the additive group of integers, the order of every elements a ≠  0 is

Test: Group Theory - 3 - Question 15

Let Z be a set of integers, then under ordinary multiplication (Z, ·) is

Test: Group Theory - 3 - Question 16

Set of all n, nth roots of unity from a finite abelian group of order n with respect to

Test: Group Theory - 3 - Question 17

The generators of a group G = {a, a2, a3, a4, a5, a6 = e) are

Detailed Solution for Test: Group Theory - 3 - Question 17

we have o(G) = 6 and prime to 6 are 1 and 5

Test: Group Theory - 3 - Question 18

If G is a group such that a2 = e, for all a ∈ G, then G is

Test: Group Theory - 3 - Question 19

Consider the system of equations x + y + z = 1, 2x + 3y + 2z = 1, 2x + 3y + (a2 – 1)z = a + 1 then

Detailed Solution for Test: Group Theory - 3 - Question 19

Given system of linear equations:
x + y + z = 1 ….(1)
2x + 3y + 2z = 1 ….(2)
2x + 3y + (a2 – 1)z = a + 1 …..(3)
Consider a2 – 1 = 2
then LHS of (2) and (3) are same but RHS are not.
Hence a2 = 3 => |a| = √3
For |a| = √3, system is inconsistence.
So option (b) is correct.

Test: Group Theory - 3 - Question 20

If G is a group, then for all a, b ∈ G

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