IIT JAM Mathematics Test Series 2023 : Group Theory – Group and its Basic Properties

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 IIT JAM Mathematics Test Series 2023 : Group Theory – Group and its Basic Properties 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.

IMG 20220804 131740 502 compress68


IIT JAM Mathematics Test Series 2023 : Group Theory – Group and its Basic Properties


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 more like e-books, video lectures, syllabus  etc regarding  your Preparation / Examination  then do 📌 mention in the Comment Section below

Hope you like the test given above for IIT JAM Mathematics 2023 of  Topic – Set Theory and Relation of the Chapter Group Theory . To get more Information about any exam, Previous Question Papers , Study Material, Book PDF, Notes etc for free, do share the post with your friends and Follow and Join us on other Platforms links are given below to get more interesting information, materials like this.

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)

Important Searches & Tags

  • IIT JAM Mock Test Papers download
  • IIT JAM test series
  • Free Mock Test mathematics Mock Test
  • IIT JAM Test series mathematics
  • Eduncle IIT JAM Test Series
  • Free Mock Test for IIT JAM Mathematics
  • IIT JAM Mathematics Mock Test 2022
  • IIT JAM Mock Test
  • IIT JAM free Mock Test 2022
  • IIT JAM Mathematics topic wise questions PDF
  • IIT JAM 2022 mock Test
  • Best Test Series for IIT JAM Mathematics
  • Free Online Test Series for IIT JAM Mathematics
  • IIT JAM mock Test 2022
  • IIT JAM free Mock Test 2022
  • Unacademy IIT JAM Mathematics test series
  • DIPS academy Test series
  • Career Endeavour Test Series IIT-JAM
  • Career Endeavour Test Series pdf

Leave a Comment

close