Mathematics Exam  >  Mathematics Tests  >  Topic-wise Tests & Solved Examples for Mathematics  >  Test: Group Theory - 9 - Mathematics MCQ

Test: Group Theory - 9 - Mathematics MCQ


Test Description

20 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Test: Group Theory - 9

Test: Group Theory - 9 for Mathematics 2024 is part of Topic-wise Tests & Solved Examples for Mathematics preparation. The Test: Group Theory - 9 questions and answers have been prepared according to the Mathematics exam syllabus.The Test: Group Theory - 9 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Group Theory - 9 below.
Solutions of Test: Group Theory - 9 questions in English are available as part of our Topic-wise Tests & Solved Examples for Mathematics for Mathematics & Test: Group Theory - 9 solutions in Hindi for Topic-wise Tests & Solved Examples for Mathematics course. Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free. Attempt Test: Group Theory - 9 | 20 questions in 60 minutes | Mock test for Mathematics preparation | Free important questions MCQ to study Topic-wise Tests & Solved Examples for Mathematics for Mathematics Exam | Download free PDF with solutions
Test: Group Theory - 9 - Question 1

 What is the maximum order of any element in A10

Test: Group Theory - 9 - Question 2

Assertion (A): A group can be isomorphic to its proper subgroup
Reason (R): The additive group Z of integer is isomorphic to (H, +) where
H = {mx : x ∈ Z and 0 ≠ m ∈ Z}

1 Crore+ students have signed up on EduRev. Have you? Download the App
Test: Group Theory - 9 - Question 3

The set of complex number z with | z | = 1 under the operation * denote  by  z1 * z2 = | z1 | ·z2

Test: Group Theory - 9 - Question 4

Let 
be the dihedral group of order 8. Then y-42 is equal to

Test: Group Theory - 9 - Question 5

 Suppose that o(a) = 24, then the a generator for  are

Detailed Solution for Test: Group Theory - 9 - Question 5

(21) = {gcd(24, 21)} = (3) = {0, 3, 6, 9, 12, 15, 18, 21}
(10) = {gcd(24, 10)} = h2i = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22}
(21) ∩ (10) = {0, 6, 12, 18} = (6)
(a21) = (agcd(24,21)) = (a3) = {e, a3, a6, a9, a12, a15, a18, a21}
(a10) = (agcd(24,10)) = (a2) = {e, a2, a4, a6, a8, a10, a12, a14, a16, a18, a20, a22}
(a21) ∩ (a10)
= (a3) ∩ (a2)
= (a6) = {e, a6, a12, a18}

Test: Group Theory - 9 - Question 6

 In an infinite cyclic group G the number of automorphism is 

Test: Group Theory - 9 - Question 7

If Gand G2 are commulator subgroup and centre of the dihedral group D4 respectively. Then select the incorrect statement.

Test: Group Theory - 9 - Question 8

U(n) is cyclic for

Detailed Solution for Test: Group Theory - 9 - Question 8

THEOREM. (Primitive Root Theorem) U(n) is cyclic if and only if n is 1, 2, 4, or 2p, where p is an odd prime and k ≥ l.

Test: Group Theory - 9 - Question 9

Let G be a group of order pq where p and q are prime numbers. Such that p > q then, G can have

Test: Group Theory - 9 - Question 10

The cardinality of the centre of D12 is

Test: Group Theory - 9 - Question 11

If in a group a5 = e, aba-1 = b2 for a,b ε G, then o(b) is 

Test: Group Theory - 9 - Question 12

Let D8 denote the group of symmetries of square (dihedral group). The minimal number of generators for D8 is

Test: Group Theory - 9 - Question 13

In the symmetric group Sn of degree n,n > 2 the set of all 3-cycles generates a subgroup of order

Detailed Solution for Test: Group Theory - 9 - Question 13


Hence the correct option is n!/2

Test: Group Theory - 9 - Question 14

The order of 2 in the field Z29 is

Detailed Solution for Test: Group Theory - 9 - Question 14

we have  25 = 32(mod 29)
implies       25 = 3(mod 29)
implies       (25)3 = (3)3 (mod 29)
implies       215 = -2(mod 29)
implies       (215)2 = (-2)2(mod 29) implies       230 = 4(mod 29)
implies       228 = 4/4 (mod 29)
implies       228= 1 (mod 29)
Hence, The order of 2 in the field z29 is 28.

Test: Group Theory - 9 - Question 15

The number of non-empty even subsets (even set is the set having even number of elements) of a set having n elements is

Detailed Solution for Test: Group Theory - 9 - Question 15

Let A - { 1 , 2 , 3 }
Then, the non-empty even subsets
nC2 + nC4 + ... = 2n-1 - 1
(even set is the set having even number of elements) of a set having three elements r { 1 , 2 } , {2 , 3 }, {1, 3}
Number of subsets = 23-1 - 1 = 3 Hence, the number of non-empty even subsets of a set having n elements is 2n-1 - 1.

Test: Group Theory - 9 - Question 16

Consider the alternating group A4 = {σ ε S4 : σ is an even permutation).
Which of the following is FALSE?

Detailed Solution for Test: Group Theory - 9 - Question 16

Given A4 = is an even permutation}
A4 has a subgroup of order 6 is false.
Because in this example converse of Lagrange’s theorem fails.
Hence, option (c) is correct.

Test: Group Theory - 9 - Question 17

Let < (0,2) > denote the subgroup generated by (0,2) in Z4 x Z8. Then the order of (3, 1) + < (0, 2) > in the quotient group Z4 x Z/ < (0,2) > is

Detailed Solution for Test: Group Theory - 9 - Question 17

Let order of <0, 2> + (3,1) is k, then k <0, 2> + (3, 1) + k(3, 1) = <0, 2>

=> Thus, k will be 4.
For this, k 4(3,1) = (12,4) = (0,4). Thus, 4 <0, 2> + (0, 4) ε <(0, 2)>

Test: Group Theory - 9 - Question 18

The order o f the quotient group Zx Z9 x Z18/ < 2 , 2 , 2 > is

Detailed Solution for Test: Group Theory - 9 - Question 18

We know that

Test: Group Theory - 9 - Question 19

Let G be a group with respect to multiplication. If x = then x-1 is

Detailed Solution for Test: Group Theory - 9 - Question 19

Given 
So,
 

Test: Group Theory - 9 - Question 20

Let G be a group of order 8 generated by a and b such that a4 = b2 = 1 and ba = a3b. The order of the center of G is

Detailed Solution for Test: Group Theory - 9 - Question 20

Let G be a group of order 8 generated by 'a' and ‘b’ s.t. a4 = b2 = 1 and ba = a3b. This group is dihedral group. (i.e. D4)
Therefore the order of the centre of G is 2.

27 docs|150 tests
Information about Test: Group Theory - 9 Page
In this test you can find the Exam questions for Test: Group Theory - 9 solved & explained in the simplest way possible. Besides giving Questions and answers for Test: Group Theory - 9, EduRev gives you an ample number of Online tests for practice
Download as PDF