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Test: Group Theory - 10 - Mathematics MCQ


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20 Questions MCQ Test - Test: Group Theory - 10

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Test: Group Theory - 10 - Question 1

The order of the permutation (12) (546) (3978) in the symmetric group S9 is

Detailed Solution for Test: Group Theory - 10 - Question 1

Order of (1 2) (5 4 6) (3 9 7 8), order of (1 2) = 2, order of (5 4 6) = 3, order of (3 9 7 8) = 4
Thus,order of composite = 1.c.m. of 2,3 and 4=12

Test: Group Theory - 10 - Question 2

Let G be a group of order 7 and  Then φ is

Detailed Solution for Test: Group Theory - 10 - Question 2

G is a group of order 7 which is prime. So group will be cyclic and each element can generate it. Now, let x generate the group, then x1 = e.

Thus, φ is one-one and onto.
Also, 

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Test: Group Theory - 10 - Question 3

The set (A x Q)\(N x N) equals

Detailed Solution for Test: Group Theory - 10 - Question 3

Test: Group Theory - 10 - Question 4

Let S1 = {2 }, S2 = {4, 6}, S3 = {8, 10,12,} S4 = {14 ,16 , 18, 20 } and so on. The sum of elements of S10 is

Detailed Solution for Test: Group Theory - 10 - Question 4

Given S1 = {2} ,
S2 = {4 , 6} ,
S3 = {8, 10, 12},
S4 = {14,16,18,20}
S10 = {9 2 , 94, 96, 98, 100, 102, 104, 106, 108, 110}
So, The sum of elements of S10 is equal to 10/2 x 202 = 1010

Test: Group Theory - 10 - Question 5

Let x * y = 3xy for all x , y ε R\{ 0 } . The inverse o f the element 2 in the group (R\{0}, *) is

Detailed Solution for Test: Group Theory - 10 - Question 5

x*y = 3xy
Let e is identity then
x*e =x implies 3xe = x implies e = 1/3
Let inverse of 2 is a then
  implies 3 x 2 x a = 1/3, So, a = 1/18. 

Test: Group Theory - 10 - Question 6

Let G be a group with identity e such that for some a ε G, a≠ e and a6 = e. Then which of the following is true?

Detailed Solution for Test: Group Theory - 10 - Question 6

Given G be a group with identity e such that for some a ε G, a2 ≠ e and a6 = e.
Hence a4 ≠ e , a≠ e .

Test: Group Theory - 10 - Question 7

Which of the following is TRUE for groups of even order?

Detailed Solution for Test: Group Theory - 10 - Question 7

If we take G = { 1 , - 1 , i, - i } then G has a non-trivial proper subgroup H = {1, - 1}
So, option (a) is discarded.
option (b) is discarded because identity is self inverse element in any group, option (c) is discarded because if we take Z6 = {0,1,2, 3,4, 5} then Z6 is a group but it is not power of 2.
For even order group apart from identity there will odd number of elements. Since inverse element exist in pairs so at least one element will be it own inverse including identity such elements will be two.

Test: Group Theory - 10 - Question 8

The number of subsets of { 1 , 2 , . . . , 10} which are disj oint from {3, 7, 8} is

Detailed Solution for Test: Group Theory - 10 - Question 8

Number of disjoint set will be chosen from remaining 7 elements.
So, number of disjoint set
= 7C0 + 7C1 + 7C2 + . . . + 7C7 = 27 = 128.

Test: Group Theory - 10 - Question 9

Let P be a set having n > 10 elements. The number of subsets of P having odd number of elements is

Detailed Solution for Test: Group Theory - 10 - Question 9

Let P be a set having n > 10 elements. The number of subsets of P having odd number of elements is nC1 + nC3 + nC5
+nC7 + nC+ nC11 = 2n-1.

Test: Group Theory - 10 - Question 10

Consider the set {1, 3, 7, 9} under the operation of multiplication modulo 10. Which one of the 4following statements about the given set is FALSE?

Detailed Solution for Test: Group Theory - 10 - Question 10

Given the set {1, 3, 7, 9} under the operation of multiplication modulo 10. i.e. U(10)= {1,3, 7, 9}
This is cyclic group because 3 and 7 are generators.
Hence, It has a unique generator is false.

Test: Group Theory - 10 - Question 11

For sets P, Q, R which of the following is NOT correct?

Detailed Solution for Test: Group Theory - 10 - Question 11

If P ∪ Q = P ∪ R then Q = R is incorrect.
For example, Let P = { 1,2 ,3 } ,Q = { 1,2 } ,R = {1} P ∪ Q = { 1 ,2 , 3} , P ∪ R = {1, 2, 3}. But Q ≠ R. 

Test: Group Theory - 10 - Question 12

Let S be a set with 10 elements. The number of subsets of S having odd number of elements is

Detailed Solution for Test: Group Theory - 10 - Question 12

Number of subset having odd number of elements
=10C1 + 10C2 + ... + 10C9 = = 512
Since ZI2 under addition modulo 12 is a cyclic group
So, Total number of non-trivial proper subgroups
= Z(12) - 2 
= 6 - 2 = 4

Test: Group Theory - 10 - Question 13

The total number of non-trivial proper subgroups of the group Z12 under addition modulo 12 is

Detailed Solution for Test: Group Theory - 10 - Question 13

F is field with 5 elements K = {{a, b )| a, b ε F}
Since, operation is component-wise thus all properties of group will be satisfied component-wise.
All elements of K will possess inverse and there is identity thus K is a field with 5 x 5 = 25 elements.

Test: Group Theory - 10 - Question 14

Let F be a field with five elements and let K={{a, b) | a, b ε F} with the binary operations defined componentwise. Then

Detailed Solution for Test: Group Theory - 10 - Question 14

We know that

where p(r) =Number of partitions of the set.

Test: Group Theory - 10 - Question 15

Let X= {1,2, 3,4}. Then the total number of partition of the set X is

Detailed Solution for Test: Group Theory - 10 - Question 15

Option (a) is incorrect 
Option (b) is incorrect because every group of finite order may or may not be cyclic.
For example K4 is finite grup but it is not cyclic.
Option (c) is correct because every cyclic group is abelian.
Option (d) is incorrect because A4 is a group and 6/o(A4)
But A4 has not a subgroup of order 6.
 

Test: Group Theory - 10 - Question 16

Which of the following is not true?

Detailed Solution for Test: Group Theory - 10 - Question 16

α = (1 3) (2 5 4)
|order of (1 3) = 2
order of (2 5 4) = 3
order of α = 1.c.m. of 2 and 3 = 6

Test: Group Theory - 10 - Question 17

If a = (13)(254) in the symmetric group S5  then α65 equals

Detailed Solution for Test: Group Theory - 10 - Question 17

So, α = 1
Thus α60 = 1
Thus α65 = α65 . α65 = α= α-1
So, inverse of a will be (4 5 2) (3 1) or (2 4 5) (3 1)
Thus (d) is the correct option.

Test: Group Theory - 10 - Question 18

For a, b ε Z, define a relation a R b if ab  ≥ 0. Then the relation R is​

Detailed Solution for Test: Group Theory - 10 - Question 18

For a, b ε z, define a relation aRb if ab ≥ 0 then the relation R is an equivalence relation.
Hence, the relation R is symmetric, reflexive and transitive.

Test: Group Theory - 10 - Question 19

Let R be a commutative ring with unity of characteristic 3, For a, b ε R, (a + b)6 is equal to

Detailed Solution for Test: Group Theory - 10 - Question 19

Given R be a commutative ring with unity of characteristics 3.
So a + a + a = 3a = 0
Hence, (a + b)6
=6C0a6 + 6C1a5b + 6C2a4b2+ 6C3a3b3+ 6C4a2b4+ 6C5ab5 + 6C6b6
=a6 + 0 + 0 + 0 + 0 + 0 + b6
= a6 + b6

Test: Group Theory - 10 - Question 20

A non empty set A is termed as an algebraic structure ________

Detailed Solution for Test: Group Theory - 10 - Question 20

 A non empty set A is called an algebraic structure w.r.t binary operation “*” if (a*b) belongs to S for all (a*b) belongs to S. Therefore “*” is closure operation on ‘A’.

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