Questions on Group and its Basic Properties with Answers

Practice Questions on Group and its Basic Properties with Answers

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Questions on Group and its Basic Properties with Answers


Q1. Let σ : {1,2,3,4,5} → {1,2,3,4,5} be a permutation (one-to-one and onto function) such that σ-1 (j) ≤ σ-1 (j) ∀j, 1 ≤ j ≤ 5. Then which of the following are true?

(a) σ•σ(j) = j for all j , 1 ≤ j ≤ 5.
(b) σ-1 (j) ≤ σ-1 for all j, 1 ≤ j ≤ 5
(c) The set {k : σ(k) ≠ k} has an even number of elements.
(d) The set {k: σ(k) = k} has an odd number of elements.


Q2. Which of the following is Non-cyclic group

(a) (pZ,+); where p is prime.
(b) (mZ,+); where m is integer.
(c) (R*,*); where R* = R – {0}
(d) None of the above.


Q3. Let C* denote the multiplicative group of non-zero complex numbers. Let G1 be the cyclic subgroup generated by 1+i and G2 be the cyclic subgroup generated by (1+i)/√2. Which one of the following is correct?

(a) Both G1 and G2 are infinite groups
(b) G1 is finite, but G2 is infinite group
(c) G1 is finite, but G2 is infinite group
(d) Both G1 and G2 are finite groups


Q4. Let σ be an element of the permutation group S5. Then the maximum possible order of σ is

(a) 5
(b) 6
(c) 10
(d) 15


Q5. Let the group G = R ( the set of all real numbers) under addition and the group H =R+ ( the set of all positive real numbers) then under multiplication.

(a) H is a cyclic group and G is a non- cyclic group.
(b) G is a cyclic group and H is a non- cyclic group
(c) Neither G nor H cyclic
(d) Both G and H are cyclic


Q6. Suppose that H is the smallest subgroup of Z under addition and H contains 18, 80, 40
then H is

(a) 18 Z
(b) 40 Z
(c) 2 Z
(d) Z


Q7. If the order of every non-identity element in a group is ‘n’, then

(a) ‘n’ is necessarily a prime number
(b) ‘n’ can be any odd number
(c) ‘n’ is an even number
(d) ‘n’ can be any positive


Q8. Let G={g1,g2,…,gn} be a finite group and suppose it is given that gi2 = identity, for i = 1,2,…,n-1. Then

(a) gn2 is identity and G is abelian
(b) gn2 is identity, but G could be non- abelian
(c) gn2 may not be identity
(d) None of the above


Q9. Consider the multiplicative group G of all the (complex)2n -th roots of unity where n = 0,1,2.. Then

(a) Every proper subgroup of G is finite.
(b) G has a finite set of generators
(c) G is cyclic
(d) Every finite subgroup of G is cyclic


Q10. Which of the following numbers can be orders of permutations σ of 11 symbols such that σ does not fix any symbol?

(a) 18
(b) 30
(c) 15
(d) 28


Q11. Let G =Z10 x Z15. Then

(a) G contains exactly one element of order 2
(b) G contains exactly 5 elements of order 3
(c) G contains exactly 24 elements of order 5
(d) G contains exactly 24 elements of order


Q12. Which of the following groups has a proper subgroup that is NOT cyclic?

(a) Z15 x Z17
(b) S3
(c) (Z,+)
(d) (Q.+)


Q13. Let G1 be an abelian group of order 6 and G2 = S3 . For j = 1,2 . let P be the statement:

“Gj has a unique subgroup of order 2″

Then


(a) both P1 and P2 hold
(b) neither P1 nor P2 hold
(c) P1 holds but not P2
(d) P2 holds but not P1


Q14. Let D8 denote the group of symmetries of square (dihedral group). The minimal number of generators for D8 is

(a) 1
(b) 2
(c) 4
(d) 8


Q15. Consider the following statements.

S: Every non abelian group has a nontrivial abelian subgroup
T: Every nontrivial abelian group has a cyclic subgroup.

Then

(a) Both S and T are false
(b) S is true and T is false
(c) T is true and S is false
(d) Both S and T are true


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

01.(a), (b), (c), (d)06.(c)11.(a), (c), (d)
02.(c)07.(a)12.(d)
03.(c)08.(a)13.(c)
04.(b)09.(a), (d)14.(b)
05.(c)10.(a), (b), (c), (d)15.(d)

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