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

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## 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

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