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Test: Group Theory - 3 - Mathematics MCQ


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20 Questions MCQ Test - Test: Group Theory - 3

Test: Group Theory - 3 for Mathematics 2024 is part of Mathematics preparation. The Test: Group Theory - 3 questions and answers have been prepared according to the Mathematics exam syllabus.The Test: Group Theory - 3 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Group Theory - 3 below.
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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

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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

Detailed Solution for Test: Group Theory - 3 - Question 8
- Conjugation Definition: In a group \( G \), element \( b \) is conjugate to \( a \) if there exists an element \( c \) in \( G \) such that \( b = c^{-1}ac \).
- Option 1 Analysis:
- \( b = c^{-1}ac \) directly matches the conjugation definition.
- It correctly shows how \( b \) is obtained by conjugating \( a \) with \( c \).
- Conclusion: Therefore, Option 1 is the correct expression for \( b \) being conjugate to \( a \).
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

Detailed Solution for Test: Group Theory - 3 - Question 14

The correct answer is:

1. infinity

In the additive group of integers, the order of an element aaa (where a≠0 ) refers to the smallest positive integer nnn such that n⋅a=0 However, for any non-zero integer aaa, there is no positive integer nnn that satisfies this equation because adding aaa to itself any number of times will never result in 0. Therefore, the order of any non-zero element in this group is infinite.

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

The value of k for which kx + 3y - k + 3 = 0 and 12x + ky = k have infinite solution is:

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



Test: Group Theory - 3 - Question 20

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

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