Practice Questions On the Fundamental of Group with Answers.

**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 Q ^{c} be the set of irrational real numbers and let G = Q^{c}U ∪{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 a ^{2} = 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}(a^{k}) > n only

(b) _{o}(a^{k}) ≥ n

(c) _{o}(a^{k}) < n only

(d) _{o}(a^{k}) ≤ 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

NOTE:- If you need anything else like e-books, video lectures, syllabus, etc. regarding your Preparation / Examination then do 📌 mention it in the Comment Section below. |

*Answer Key*

*Answer Key*

01. | (a), (c) | 06. | (c) | 11. | (d) |

02. | (d) | 07. | (d) | 12. | (d) |

03. | (c) | 08. | (a) | 13. | (c) |

04. | (b) | 09. | (d) | 14. | (a), (d) |

05. | (c) | 10. | (d) | 15. | (d) |