Questions On Fundamental of Group with Answers

Practice Questions On the Fundamental of Group with Answers.

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Questions On Fundamental of Group with Answers

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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Answer Key

01.(a), (c)06.(c)11.(d)
04.(b)09.(d)14.(a), (d)

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