# IIT JAM Mathematics Test Series 2023 : Group Theory – Fundamental of Group

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic Fundamental of Group of the Chapter Group Theory. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Share with Your Friends.

## IIT JAM Mathematics Test Series 2023 : Group Theory – Fundamental of Group

Q1. If x,y and z are elements of a group such that xyz =1, then

(a) yzx = 1.
(b) yxz =l.
(c) zxy =1.
(d) zyx =1.

Q2. Which of the following is a subgroup of (C, +).

(a) (R, +)
(b) (G, +), where G = {πr | r ∈ Q}
(c) (G, +), where G = {ir | r ∈ R}
(d) (G, +), where G= {πn | n ∈ Z}

Q3. The value of a for which G = {a,1,3,9,19,27} is a cyclic group under multiplication modulo 56 is

(a) 5
(b) 15
(c) 25
(d) 35

Q4. Let U(n) be the set of all positive integers less thann and relatively prime to n. Then U(n) is a ground under multiplication modulo n. For n = 248, the number of elements in U(n) is

(a) 60
(b) 120
(c) 180
(d) 240

Q5. Let Qc be the set of irrational real numbers and let G = QcU ∪{0}. Then, under the usual addition of real numbers, G is

(a) A group, since R and Q are groups under addition
(b) A group, since the additive identity is in G
(c) Not a group, since addition on G is nota binary operation
(d) Not a group, since not all elements in G have an inverse

Q6. Let G be a group such that a2 = e for each a∈G, where e is the identity element of G .Then

(a) G is cyclic
(b) G is finite
(c) G is abelian
(d) None of these

Q7. In the group {1,2,…,16} under the operation of multiplication modulo 17, the order of the element 3 is

(a) 4
(b) 8
(c) 12
(d) 16

Q8. On Z+ , define * by a*b = c, where c is at least 5 more than a + b then,

(a) * is not a binary operation
(b) * is non-commutative binary operation
(c) * is commutative binary operation
(d) * is associative binary operation

Q9. Let G = { a ∈ R : a > 0, a ≠ 1} , define a*b = a logb then

(a) (G,*) is semi group but not a group
(b) (G,*) is a monoid, but not a group
(c) (G,*) is a group
(d) (G,*) is an abelian group

Q10. The set of real numbers is a group with respect to

(a.) Arithmetic subtraction
(b.) Arithmetic multiplication
(c.) Arithmetic division
(d.) Composition defined by a•b = a + b + 1 for all real a and b

Q11. Let G be a group and let a ∈ G if o(a) = n and k is any integer. Then which one of the following is correct ?

(a) o(ak) > n only
(b) o(ak) ≥ n
(c) o(ak) < n only
(d) o(ak) ≤ n

Q12. In a set R of real numbers, * be defined as a*b = a + 2b then * is

(a) Commutative
(b) Associative
(c) Not a Binary Operation
(d) Not associative but binary operations

Q13. Consider the following statements in respect of a finite group G:

A. O(a) = O(a-1) for all a ∈ G
B. O(a) = O(bab-1) for all a, b ∈ G

Which of the statements given above is/are correct?

(a) A only
(b) B only
(c) Both A and B
(d) Neither A nor B

Q14. In the set Q of rational numbers defined * as follows: for α, β ∈ Q, α*β = (α.β)/3 . If Q+, Q, Q* respectively denote the sets of positive, negative and non-zero ratioanls, the which one of the following pairs is an abelian group?

(a) (Q+,*)
(b) (Q,*)
(c) (Q,*)
(d) (Q*,*)

Q15. Let M (R) be set of all matrices with real entries. The usual matrix addition +” is

(a) Commutative binary operation
(b) Non-commutative binary operation
(c) Associative binary operation
(d) Not a binary operation

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